AN ABSTRACT OF THE DISSERTATION OF
Nicolai Thum for the degree of Doctor of Philosophy in Atmospheric Sciences
presented on September 13, 2006.
Title: Atmospheric Boundary Layer Coupling to Midlatitude
Mesoscale Sea Surface Temperature Anomalies
Abstract approved:
Steven K. Esbensen
This thesis examines the mechanisms that couple the monthly-averaged
atmospheric boundary layer (ABL) to open-ocean sea surface temperature (SST)
perturbations on scales of 50-500 km. The observed positive correlation between
surface wind speed anomalies and SST anomalies is successfully simulated using
the Weather Research and Forecasting (WRF) model.
In numerical experiments with idealized SST fronts, the cross-frontal surface wind acceleration in the cold-to-warm case and deceleration in the warmto-cold case are found over narrow transition zones co-located with the narrow
regions of large frontal SST changes. In the transition zone, horizontal momentum is redistributed vertically in the ABL by turbulence and convection. The
largest pressure adjustments, on the other hand, take place over a much broader
region downstream from the SST front. In the cold-to-warm transition zone the
model simulates an unstable thermal internal boundary layer (TIBL) in the lower
part of the ABL. As the TIBL grows, higher velocity air aloft is incorporated
into the TIBL, accelerating the flow. Over the warm-to-cold transition zone, the
momentum boundary layer collapses, and vertical mixing of momentum by turbulence and convection ceases in the upper part of the ABL.
The WRF model is also applied to open-ocean ABL flow across idealized
sinusoidal SST anomalies having scales similar to those observed in the Agulhas
return current region. The simulated horizontal pressure gradient force anomalies
are crucial to the response over the entire domain, and the vertically integrated
momentum budget is found to be approximately linear. A linear diagnostic model
is therefore developed which successfully predicts the observed phase and amplitude of the ABL wind, pressure and temperature response to the SST anomalies,
with largest quantitative discrepancies found in the perturbation wind component
perpendicular to the mean wind direction. By using the divergence and vorticity
budgets, the diagnostic model shows that differences in the vertical structure of
the perturbation wind components down and across the mean wind can explain
the differences in the coupling coefficients for the divergence and curl as functions
of downwind and crosswind SST gradients, and as functions of the angle between
the SST gradient and the mean wind.
c
Copyright by Nicolai Thum
September 13, 2006
All Rights Reserved
Atmospheric Boundary Layer Coupling to Midlatitude Mesoscale Sea Surface
Temperature Anomalies
by
Nicolai Thum
A DISSERTATION
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
Presented September 13, 2006
Commencement June 2007
Doctor of Philosophy
dissertation
of
Nicolai Thum
presented
on
September 13, 2006
APPROVED:
Major Professor, representing Atmospheric Sciences
Dean of the College of Oceanic and Atmospheric Sciences
Dean of the Graduate School
I understand that my dissertation will become part of the permanent collection
of Oregon State University libraries. My signature below authorizes release of my
dissertation to any reader upon request.
Nicolai Thum, Author
ACKNOWLEDGMENT
Foremost, I would like to express my gratitude to my advisor Steve
Esbensen for providing me the opportunity to become his student after I participated in an exchange program between Oregon State University and the University of Karlsruhe. His profound scientific knowledge, time for many meetings
and discussions and editorial advice are invaluable for this dissertation.
I would like to thank Dudley Chelton for all his help and inspiring enthusiasm for the period of my graduate studies.
I also would like to acknowledge Eric Maloney for providing the aquaplanet boundary conditions, Eric Skyllingstad for granting access to the highperformance computing facilities and Larry O’Neill for providing QuikSCAT and
AMSR satellite data and performing related calculations.
This material is based upon work supported by the National Science Foundation under Grand No. 0002322-ATM.
The non-scientific support was equally important during the time of this dissertation. I am grateful for my parents Margret and Herbert Thum for their love,
trust and support. I thank my sister Corina for sending good thought across the
ocean and all my friends for the good times, outside activities, many laughs and
company.
I am deeply grateful for my wife Annette for all her love, support and her
believe in me.
TABLE OF CONTENTS
Page
1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
WRF SIMULATION OF SURFACE WIND RESPONSE
TO AGULHAS CURRENT SST ANOMALIES . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.1
8
WRF-model description and boundary layer parameterization. . . . . .
2.1.1 Special consideration of turbulence and entrainment . . . . . . . 13
2.2
Comparison of simulated and observed wind response . . . . . . . . . . . . . . 16
2.2.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3
ABL RESPONSE TO MIDLATITUDE SST FRONTS . . . . . . . . . . . . . . . . . . 30
3.1
Simulations with idealized SST fronts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.1.1 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.2 Analysis of the simulated momentum budget . . . . . . . . . . . . 43
3.2
Results from cold-to-warm SST front experiment (P4) . . . . . . . . . . . . . 44
3.2.1 Equilibrium and quasi-equilibrium region (P4) . . . . . . . . . . . 45
3.2.2 Non-equilibrium transition region (P4) . . . . . . . . . . . . . . . . . 47
3.2.3 ABL average and surface momentum budget (P4) . . . . . . . . 48
3.3
Results from warm-to-cold SST front experiment (M4). . . . . . . . . . . . . 50
3.3.1 Equilibrium and quasi-equilibrium region (M4) . . . . . . . . . . . 50
3.3.2 Non-equilibrium advection region (M4) . . . . . . . . . . . . . . . . . 52
3.3.3 ABL average and surface momentum budget (M4) . . . . . . . . 53
4
3.4
Comparison of the responses between P4 and M4 . . . . . . . . . . . . . . . . . . 54
3.5
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
ABL RESPONSE TO IDEALIZED MESOSCALE
MIDLATITUDE SST ANOMALIES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
TABLE OF CONTENTS (Continued)
Page
4.1
Model description and initial fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.1.1 SST Forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
4.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.2
5
6
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
A LINEAR DIAGNOSTIC MODEL OF ABL RESPONSE TO
MESOSCALE SST ANOMALIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1
A Linearized ABL-averaged diagnostic model . . . . . . . . . . . . . . . . . . . . . . 111
5.2
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
APPENDIX A WRF-model η-levels and nominal heights . . . . . . . . . . . . . . . . 140
APPENDIX B Solution to the Diagnostic Model Appendix . . . . . . . . . . . . . . 141
LIST OF FIGURES
Figure
Page
2.1
Map of averaged July 2002 AMSR SST. Shaded SST contours are
shown in 2◦ C intervals. Gray rectangles denote the boundaries of the
WRF simulation. Missing SST data around islands are masked white
and interpolated for the aqua-planet simulation. SST contours are
used as overlays in subsequent map figures. . . . . . . . . . . . . . . . . . . . . 22
2.2
Comparison between WRF simulation (top) and QuikSCAT observation (bottom) averaged over the month of July 2002. Maps show
inner WRF domain simulations and corresponding QuikSCAT observations of wind speed perturbations. Contours of SST perturbation
are overlaid. The contour interval is 1◦ C. . . . . . . . . . . . . . . . . . . . . . . 23
2.3
As in Fig. 2.2, except for zonal wind. SST contour interval is 2◦ C. . . 24
2.4
As in Fig. 2.2, except for meridional wind. SST contour interval is
2◦ C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.5
Binned scatterplots of the relationships between the perturbation divergence and the perturbation downwind SST gradient: (top) perturbation divergence of WRF-model and (bottom) perturbation divergence of QuikSCAT observation. The points are the means within
each bin computed from 28 consecutive days, and the error bars are
±1 std dev of the means within each bin. The lines through the points
represent least square fits of the binned overall means to straight lines. 26
2.6
As in Fig. 2.5, except for the perturbation curl and the perturbation
crosswind SST gradient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.7
Binned scatterplots of the angular relationships between the wind and
the SST field. Top panel shows WRF perturbation wind divergence
as a function of the angle between crosswind and downwind SST gradient. Bottom panel shows QuikSCAT perturbation wind divergence
as function of the angle between crosswind and downwind SST gradient. The points are the means within each bin computed from 28
consecutive days, and the error bars are ±1 std dev of the means
within each bin. The lines through the points represent least square
fits of the binned overall means to a cosine curve. . . . . . . . . . . . . . . . 28
2.8
As in Fig. 2.7, except for the WRF perturbation curl (top) and the
QuikSCATperturbation curl as a function of the angle between crosswind and downwind SST gradient. The lines through the points represent least square fits of the binned overall means to sine. . . . . . . . . . 29
LIST OF FIGURES (Continued)
Figure
Page
3.1
Maps of background SST in the region of the Agulhas return current.
Shaded SST is shown in 2◦ C contour intervals. Gray rectangles denote the boundaries of the 3 nested WRF domains. Inside the WRF
domains the SST is increased by 4 K within a narrow north-south
frontal region. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.2
Bin-averaged distribution of normalized momentum boundary layer
depth as function of longitude in the upstream region. Each vertical
profile is a distribution at a given longitude. A thick line denotes the
ABL depth that occurred most often. Top panel is derived from the
P4 experiment and the bottom panel from the M4 experiment. The
vertical bin size for each distribution is 50m. Shown is the region upstream of the SST front where the momentum boundary layer depth
is not influenced by the SST front. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3
Longitude-height cross-section of the potential temperature θ. Top
panel shows results from P4 and bottom panel results from M4. The
solid line in each panel denotes the momentum boundary layer depth;
the thick dashed line in the upper panel is a measure of the internal
boundary layer depth. See text for the definitions of the boundary
layer depths. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4
Sea surface temperature at 43◦ S as function of longitude. Dots represents model gridpoint locations. Latitudinal dependence of SST is
approximately a cosine function of the latitude. . . . . . . . . . . . . . . . . . 62
3.5
Top: Near-surface advection as a function of longitude. Bottom: Colocated sea surface temperature distribution as function of longitude.
63
3.6
Schematic description of the cold-to-warm experiment. . . . . . . . . . . . 64
3.7
As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
velocity of the P4 experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.8
As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
turbulent stress convergence of the P4 experiment. . . . . . . . . . . . . . . 66
3.9
As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
turbulent stress of the P4 experiment. . . . . . . . . . . . . . . . . . . . . . . . . 67
3.10 As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
advection of the P4 experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
LIST OF FIGURES (Continued)
Figure
Page
3.11 As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
pressure gradient force of the P4 experiment. . . . . . . . . . . . . . . . . . . . 69
3.12 As in Fig. 3.3, except for the turbulent kinetic energy of the P4 experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.13 As in Fig. 3.3, except for the buoyancy production of turbulent kinetic
energy of the P4 experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.14 Ratio of surface stress to momentum boundary layer depth as function
of longitude. Solid line denotes the magnitude τs /H, dashed the zonal
component τx /H and dotted the meridional component τy /H. . . . . . . 72
3.15 Momentum boundary layer average zonal (top) and meridional (bottom) momentum budget as function of longitude. The zonal and the
meridional residual is denoted by X, Y, respectively. . . . . . . . . . . . . . 73
3.16 Zonal (top) and meridional (bottom) momentum balance at near surface model level (12m) as function of longitude. Shown are advection
(solid), coriolis (thick dash-dotted), pressure gradient (dashed) and
turbulent stress convergence (thick dotted). . . . . . . . . . . . . . . . . . . . . 74
3.17 Vertically staggered vector representation of the momentum budget
from P4. Northward vectors are pointing up and eastward vectors
point to the right. Top panel vectors represent Coriolis (orange), advection (blue), turbulent stress convergence (“τ ”, black), and pressure
gradient force (“PGF”, green). Additionally wind direction (“Wind”)
is overlaid in gray. Bottom panel shows selection of top panel magnified with colored vectors for advection (blue) and turbulent stress
convergence (black), other vectors are repeated for completeness, but
grayed-out for clarity. Note that not all grid locations are showed. . . 75
3.18 Top: Near-surface advection as a function of longitude. Bottom: Colocated sea surface temperature distribution as function of longitude.
76
3.19 As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
velocity for the M4 experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
3.20 As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
turbulent stress convergence of the M4 experiment. . . . . . . . . . . . . . . 78
3.21 As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
turbulent stress of the M4 experiment. . . . . . . . . . . . . . . . . . . . . . . . . 79
LIST OF FIGURES (Continued)
Figure
Page
3.22 As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
advection of the M4 experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.23 As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
pressure gradient force of the M4 experiment. . . . . . . . . . . . . . . . . . . 81
3.24 As in Fig. 3.3, except for the turbulent kinetic energy of the M4
experiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.25 Ratio of surface stress to ABL depth as function of longitude. The
solid line denotes the magnitude τs /H, dashed the zonal component
τx /H and dotted the meridional component τy /H. . . . . . . . . . . . . . . . 83
3.26 Momentum boundary layer average zonal (top) and meridional (bottom) momentum budget as function of longitude. The zonal and the
meridional residual is denoted by X, Y, respectively. . . . . . . . . . . . . . 84
3.27 Zonal (top) and meridional (bottom) Momentum Balance at near surface model level (12m) as function of Longitude. Shown are Advection
(solid), Coriolis (thick dash-dotted), Pressure gradient (dashed) and
turbulent stress convergence (thick dotted). . . . . . . . . . . . . . . . . . . . . 85
3.28 Vertically staggered vector representation of the momentum budget
from M4. Northward vectors are pointing up and eastward vectors
point to the right.Top panel vectors represent Coriolis (orange), advection (blue), turbulent stress convergence (“τ ”, black), and pressure
gradient force (“PGF”, green). Additionally wind direction (“Wind”)
is overlaid in gray. Bottom panel shows selection of top panel magnified with colored vectors for advection (blue) and turbulent stress
convergence (black), other vectors are repeated for completeness, but
grayed-out for clarity. Note that not all grid locations are shown. . . . 86
3.29 Longitude-height cross section of 1) zonal turbulent flux convergence
from P4 and M4 added and zonal mean removed (top) and 2) zonal
advection from P4 and M4 added and zonal mean removed (bottom).
87
3.30 Vertical profile of the zonal (top) and meridional (bottom) momentum
budget from P4 at three locations. Shown from left to right are the
upstream, transition and downstream region relative to the SST front.
Note the order-1 difference of scales of zonal and meridional budgets.
88
LIST OF FIGURES (Continued)
Figure
3.31 Vertical profile of the zonal (top) and meridional (bottom) momentum
budget from M4 at three locations. Shown from left to right are the
upstream, transition and downstream region relative to the SST front.
Note the order-1 difference of scales of zonal and meridional budgets.
Page
89
4.1
Maps of background SST in the region of the Agulhas return current.
Shaded SST is shown in 2◦ C contour intervals. Gray rectangles denote
the boundaries of the 2 nested WRF domains. Contour interval for
the WRF domains is 1◦ C, outside contour interval 2◦ C . . . . . . . . . . . . 101
4.2
Map of near-surface zonal (top) and meridional (bottom) wind speed
perturbations with SST contour and mean wind speed vector overlay. 102
4.3
Composites of boundary layer variables as function of longitude: SST
perturbation and boundary layer depth (top panel) near the location
of maximum boundary layer depth variation. Perturbation of advection and turbulent stress convergence (second panel). Ratio of perturbation surface stress and boundary layer depth (“τ /H”), Coriolis
force (“Coriolis”) and pressure gradient force (“PGF”) (third panel).
Longitude-height cross section (bottom panel) of meridional turbulent
stress convergence (shaded) with zonal velocity (black) and boundary
layer depth (gray) contours overlaid. . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4
Composites of boundary layer variables as function of longitude: SST
perturbation and boundary layer depth (top panel) near the location of maximum boundary layer depth variation. Perturbation of
advection and turbulent stress convergence (second panel). Coriolis
force (“Coriolis”) and pressure gradient force (“PGF”) (third panel).
Longitude-height cross section (bottom panel) of meridional turbulent stress convergence (shaded) with meridional velocity (black) and
boundary layer depth (gray) contours overlaid. . . . . . . . . . . . . . . . . . 104
4.5
Composites of surface pressure gradient force and SST. Top panel
shows zonal and bottom panel meridional maps of shaded “PGF”
and contour SST overlaid. Zero contour lines are omitted for clarity.
4.6
105
Composites of surface pressure gradient force and SST. Top panel
shows modeled PGF from the WRF simulation and bottom panel
PGF estimates based on the horizontal change of potential temperature θ (Eqn. 4.3). Contours of SST are overlaid. Zero contour lines
are omitted for clarity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
LIST OF FIGURES (Continued)
Figure
Page
4.7
Boundary layer integral of zonal PGF and SST. The composite of the
top row is identical to the bottom row shifted by one-half wavelength
to show the spatial structure. Note that some additional smoothing
has been applied to remove very high wavenumber noise of unknown
origin. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
4.8
As in (Fig. 4.7, except for the zonal stress. . . . . . . . . . . . . . . . . . . . . . 108
4.9
As in (Fig. 4.7, except for the zonal advection. . . . . . . . . . . . . . . . . . . 109
4.10 As in (Fig. 4.7, except for the zonal momentum budget. . . . . . . . . . . . 110
5.1
Latitude-height cross section of estimated pressure gradient force with
ABL depth (bottom panel) and SST (top panel). Estimate of “PGF”
is shaded with contour of ABL depth (gray) as overlay. . . . . . . . . . . . 123
5.2
Map of 1-layer diagnostic model zonal velocity (top) and WRF-model
averaged zonal surface velocity (repeated for comparison, bottom).
SST contours are shown as overlays with contour interval of .5◦ C.
Zero contour of SST is omitted for clarity. . . . . . . . . . . . . . . . . . . . . . . 124
5.3
Map of 1-layer diagnostic model meridional velocity (top) and WRFmodel averaged meridional surface velocity (repeated for comparison,
bottom). SST contours are shown as overlays with contour interval
of .5◦ C. Zero contour of SST is omitted for clarity. . . . . . . . . . . . . . . . 125
5.4
Map of surface to vertically integrated meridional velocity ratio derived from WRF-model simulation. Top panel shows zonal velocity ratio, bottom panel shows meridional velocity ratio. Locations
with vertically integrated velocities of less than 0.05 m/s have been
masked. SST contours are shown as overlays with contour interval of
.5◦ C. Positive values of SST is shown as solid and negative SST as
dashed contours, the zero contour of SST is omitted for clarity. . . . . . 126
5.5
Map of 1-layer diagnostic model divergence and SST perturbations. . . 127
5.6
Map of 1-layer diagnostic model curl and SST perturbations. . . . . . . . 128
5.7
Binned scatterplots of the relationships between the SST and wind
velocity fields calculated from the diagnostic solution of the 1-layer
model. Top panel shows perturbation wind divergence as a function of
downwind SST gradient and bottom panel shows perturbation wind
curl plotted as a function of crosswind SST gradient. . . . . . . . . . . . . 129
LIST OF FIGURES (Continued)
Figure
5.8
Page
Binned scatterplots of the angular relationships between the wind
and the SST field. Top panel shows perturbation wind divergence as
a function of the angle between down and crosswind SST gradient
and bottom panel shows perturbation wind curl plotted as a function
of the angle between down and crosswind SST gradient. . . . . . . . . . . 130
LIST OF TABLES
Table
5.1
Page
Parameters for the 1-layer linear perturbation model. The first 11 parameters were pre-determined by the experimental design and physics.
The last three values represent model parameters with a range of
physically correct values (see text for details). For these three values the optimal value within the physical range were chosen. The
sensitivity of the model is indicated and the influence effect is given.
122
Atmospheric Boundary Layer Coupling to Midlatitude Mesoscale Sea
Surface Temperature Anomalies
1. INTRODUCTION
This dissertation focuses on monthly-averaged atmospheric boundary layer
(ABL) response to midlatitude sea surface temperature (SST) anomalies on scales
of 50-500 km. The oceanic mesoscale corresponds roughly to the atmospheric
mesoscale β and the designation β will be dropped for simplicity. Near-surface
wind speed anomalies on these scales have been observed to be co-located with
SST anomalies. These correlations have only recently been revealed thanks to the
advances in microwave satellite technology which have enabled high resolution
surface wind stress and SST observations in virtually all-weather conditions.
Important processes that affect the near-surface winds on the scale of the
SST anomalies include turbulent vertical mixing, horizontal momentum advection,
turbulent entrainment, and the horizontal surface pressure gradient force. In
general, all of these processes play a role in the ABL response. There has been,
however, much speculation about the specific mechanism coupling the wind stress
to the SST.
The quantitative theories proposed to date may be grouped into three
categories:
• response of ABL wind to thermally induced changes in pressure gradient
forcing (e. g. Deser 1993; Small et al. 2005)
2
• downward turbulent mixing of momentum with relatively constant pressure
gradient forcing (e. g. de Szoeke and Bretherton 2004).
• changes in surface stress due to variations in boundary layer depth under
relatively constant pressure gradient forcing (e. g. Samelson et al. 2006).
Here we perform mesoscale ABL model simulations with the necessary
horizontal and vertical resolution to resolve important boundary layer processes.
A detailed analysis of the simulated momentum budget is used to identify scale
dependent mechanisms of the ABL adjustment and to determine which theory
comes closest to explaining the observations over the midlatitude open oceans.
The numerical model is validated by comparison of the simulated nearsurface winds and satellite near-surface wind observations over SST anomalies
of the Agulhas current system. We then carry out numerical experiments on
the ABL response to idealized SST-fronts and mesoscale SST anomalies similar
in magnitude and scale to the observed SST anomalies of the Agulhas region
(Fig. 3.4, Fig. 3.18, and Fig. 4.1).
A more complete literature review will be presented in the following chapters, but it is useful to mention the seminal ideas at this point that have guided
much of the work on mesoscale coupling mechanisms. Suggestions regarding the
importance of thermodynamic processes can be traced to the work of Lindzen
and Nigam (1987). They suggest that surface heat flux over the warm tropical
SST increases the lower-tropospheric air temperature. Hydrostatic adjustment
then leads to surface pressure gradients with patterns of wind convergence over
the warm SST and divergence over cool SST. The wind response is strongest over
the strongest SST gradient. On the mesoscale, Small et al. (2005) argue that
temperature advection shifts the strongest surface heat fluxes downstream of the
3
SST-front, for example on the north side cold of the tongue in the eastern equatorial Pacific. The shift of increased air temperature corresponds to increased
surface pressure gradients downstream of the SST-front. The pressure gradient
is therefore not centered over the front but shifted downstream and surface wind
acceleration occurs on the warm downstream side of the SST-front. Note that
the horizontal wind stress plays a passive role in the thermodynamic mechanism.
The stress convergence and coriolis effects are just what is needed to balance the
thermodynamically driven pressure gradients in the momentum budget.
The second class of mechanisms proposes that SST anomalies induce
changes in the turbulent momentum fluxes and drive the adjustment mechanism.
These ideas can be traced to the observational work of Sweet et al. (1981) and
Wallace et al. (1989). They suggest that the SST induced changes of ABL stability
alter the turbulent vertical mixing of momentum to the surface. Over cold SST
the boundary layer tends to be more stable and the turbulent momentum flux to
the surface is reduced; over warm SST mixing reduces the vertical shear and leads
to increased surface wind. The quasi-Lagrangian large eddy simulations of the
eastern tropical Pacific with prescribed pressure gradient force by de Szoeke and
Bretherton (2005) support the hypothesized importance of the turbulent mixing
effect, but were unable sort out the relative importance of wind stress convergence
and pressure adjustment as ABL response mechanisms.
The potential importance of ABL depth in determining the equilibrium
ABL response was recently demonstrated by Samelson et al. (2006) using an idealized quasi 2-dimensional 2-layer model. The modeled horizontal pressure gradient
was held constant in time and space while the depth of the momentum boundary layer was allowed to change across the SST-front. Under these conditions, a
deeper boundary layer over warm water can sustain larger surface stress than a
4
shallow ABL over cooler water. The ratio of ABL depth and surface stress must
remain constant across the SST-front and the surface stress in this 2-layer model
is founded to be independent of the momentum distribution within the boundary
layer.
In this thesis we will analyze numerical simulations of the ABL adjustment
to SST-fronts and mesoscale SST anomalies to sort out the relevant mechanism on
meso- to synoptic scales. In Chapter 2 we will test the ability of the Weather Research and Forecasting (WRF) model, including the Grenier and Bretherton (2001,
hereafter GB01) moist PBL-scheme, to simulate the observed wind-SST coupling.
The WRF-model is a modern, fully modularized mesoscale numerical weather prediction system and is becoming the successor of the widely adopted Pennsylvania State University/National Center for Atmospheric Research mesoscale model
(known as MM5). We show that the WRF-model can simulate the correct location and the correct strength of the divergence and convergence pattern relative
to the SST gradients. Momentum advection is shown to play an important role
in the adjustment of the boundary layer and is fundamental to reproducing the
SST-divergence pattern accurately. The method of Chelton et al. (2001) is used
to calculate coupling coefficients for wind divergence perturbation as a function
of SST gradient perturbations (see also Chelton et al. 2004; O’Neill et al. 2003;
2005). These results are used to quantify the relationship of the model divergence
to SST gradients and to compare the simulated ABL response with satellite observations. Chapter 2 also provides an introduction to the GB01 moist PBL-scheme
and discusses the general WRF-model setup and provide. A literature review
focuses on the modeling aspect of previous studies.
In Chapter 3 the WRF-GB01 model is used to simulate the response of
the boundary layer to idealized step-like SST fronts. The chapter provides a more
5
detailed literature review on the suggested mechanisms of ABL adjustment. The
design of the SST-front experiments were inspired by the analogy to the method
of determining the response of a linear system to step-like excitation. Although
the finite difference scheme of the WRF-model does not allow for true step-like
SST changes, the imposed SST gradient is comparable to large observed SST
gradients in the open ocean (de Szoeke et al. 2005). Two independent model runs
were performed to simulate the boundary layer response to a positive SST jump
and a negative SST jump in the direction of the mean background wind, thereby
simulating a warm-to-cold and a cold-to-warm transition. The strongest model
responses were found on small scales of less than 200 km which give rise to the
co-located pattern of surface wind divergence and SST-gradient. The responses
from cold-to-warm and from warm-to-cold are not symmetric, i. e., not strictly
linear, and by means of the momentum budget and the internal structure of the
boundary layer we discuss and isolate the dominant physical mechanisms in the
case of warm and cold frontal transitions.
In Chapter 4 we apply our understanding of the boundary layer response
from Chapter 3 to simulations of wind response to simplified SST anomalies that
are more characteristic of the open ocean. In the Agulhas region, the monthly
averaged SST pattern (O’Neill et al. 2005) appears to be a quasi-periodic series of
warm and cold anomalies with spatial scales on the the order of 200-500 km giving
ABL horizontal advective timescales of ∼ 6 hours. These scales suggest that the
ABL must continually adjust to changing SST over the entire Agulhas region.
The perturbations of the pressure gradient and Coriolis force are found to be
important on the mesoscale and we speculate about ABL adjustment mechanisms
as a function of scale.
6
Chapter 5 presents a simple 1-layer diagnostic ABL model for mesoscale
perturbations of purely zonal mean flow. The 1-layer model results are in good
agreement with the vertically averaged WRF-GB01 results in Chapter 4 and allow
for prediction of phase shifts and magnitude of the perturbation wind and wind
stress in terms of SST phase and magnitude. The coupling coefficients first used
by Chelton et al. (2001) to quantify the relation between SST and surface wind
stress are well predicted by the 1-layer model and it shows that the driving force
for the ABL-averaged flow is the pressure gradient term.
7
2. WRF SIMULATION OF SURFACE WIND RESPONSE
TO AGULHAS CURRENT SST ANOMALIES
Advances in microwave satellite technology allow an unobstructed view of
the world’s ocean because clouds are virtually transparent in the microwave band
except during periods of heavy precipitation. Newly available high resolution
satellite observation of near-surface wind and sea surface temperature (SST) have
increased the knowledge about boundary layer adjustments over changing SST.
The coupling of the divergence and the curl of the wind field to the gradient
of SST is well established. However, the two dimensional satellite observations
at the ocean surface alone only allow speculations about the manifestation of
surface properties throughout the atmospheric boundary layer (ABL). The lack
of detailed three dimensional observations over most of the global ocean makes
numerical models an attractive tool for the study of air-sea interactions. Previous
numerical simulations, however, have not lead to any consensus on the mechanism
that connects mesoscale SST anomalies with the low-level winds.
The primary goal of this chapter is to introduce and validate the Weather
Research and Forecast (WRF) model for our purposes and to explain the basic experimental setup for a realistic ABL simulation over the Agulhas region.
The chapter will also be used to describe the initial and background state of the
idealized experiments of Chapter 3.
This chapter will first provide an introduction to the WRF-model and the
boundary layer scheme we have implemented in WRF to improve the quality and
physical interpretation of the simulations. The WRF-model is applied in a realistic
simulation of the boundary layer response over Agulhas return current where high
resolution satellite observations of SST and near surface winds allow quantitative
comparison with observations.
8
2.1. WRF-model description and boundary layer parameterization
We use the Weather Research and Forecasting (WRF) modeling system
version 2 (Skamarock et al. 2005) with a state-of-the-art moist planetary boundary layer scheme (Grenier and Bretherton 2001, hereafter GB01). The key component of the WRF-model is the Advanced Research WRF (ARW) dynamic solver
(Wang et al. 2006). The solver provides the solutions to the fully compressible
nonhydrostatic equations on a mass-based terrain-following coordinate system.
The WRF-model provides two-way nesting capabilities and full physics options
for land-surface, boundary layer, radiation, microphysics and cumulus parameterization.
In this section we briefly describe all components of the model. The WRF
software framework provides the modular infrastructure for ease in combining
various representations of physical processes in the atmosphere. It is a practical
impossibility to test all available model configurations. Whenever possible we
choose physics options that performed well in previous studies or are the defaults
for fifth generation Pennsylvania State University–NCAR mesoscale model (MM5)
and the NCEP Eta-model. The microphysics includes resolved water vapor, cloud,
and precipitation processes. In this study the simple WRF Single-Moment 3-class
(WSM3) scheme is used and is carried out at the end of the time step. This guarantees that the final saturation balance is accurate for the updated temperature
and moisture (Skamarock et al. 2005). Atmospheric radiative heating and radiative flux divergence were calculated using the Rapid Radiative Transfer Model
Longwave parameterization (Mlawer et al.1997) and the Eta Geophysical Fluid
Dynamics Laboratory Shortwave parameterization (Lacis and Hansen 1974). The
radiation physics package is only called every 4th time step (12 min.) to reduce the
9
computation. Radiation does not change significantly in this time. The modified
version of the Kain-Fritsch scheme (Kain and Fritsch 1990; Kain 2004) is used to
represent sub-grid-scale effects of convection and shallow clouds. In terms of cumulus physics, the grid size of 8.3 km is fine enough to resolve some features of the
largest convective eddies, however, the convective eddies are not entirely resolved
and the cumulus parameterization scheme must represent the remaining effects
of cumulus convection on smaller spatial scales (Skamarock et al. 2005), such ass
releasing latent heat in the convective column. In addition this scheme provides
the heating and moistening effects from shallow cumulus. In internally stratified
boundary layers the effects of shallow clouds become important for the ABL parameterization because of their effects on the turbulent kinetic energy budget and
hence the mixing coefficients. The surface layer scheme provides surface exchange
coefficients for heat, moisture, and momentum. It does not calculate any tendencies, but only provides stability-dependent information to the land-surface and
the ABL scheme. We chose the similarity-theory-based MM5 scheme which uses
a Charnock relation to relate roughness length to friction velocity over the ocean.
The calculation of a complex surface energy budget is not necessary because we
prescribe the SST and thus we were able to use a simple thermal diffusion landsurface scheme. It provides sensible and latent heat flux to the ABL scheme. We
completed the implementation by McCaa (personal communication) of the GB01
moist ABL parameterization. The details of the GB01 ABL scheme are discussed
below.
A time-splitting integration scheme is used to solve the Euler equations.
The basic set of equation is given below in their flux-forms:
10
∂t U + (∇ · Vu)η − ∂x (pφη ) + ∂x (pφx ) = FU
(2.1)
∂t V + (∇ · Vv)η − ∂y (pφη ) + ∂y (pφy ) = FV
(2.2)
∂t W + (∇ · Vw)η − g(∂y p − η) = FW
(2.3)
∂t Θ + (∇ · Vθ)η = FΘ
(2.4)
∂t µ + (∇ · V)η = 0
h
i
∂t φ + µ−1 (V · ∇φ)η − gW = 0
(2.5)
(2.6)
with the vertical coordinate µ and the flux form variables defined as
µ = (ph − pht )/µ where µ = phs − pht
V = µv = (U, V, W ), v = (u, v, w), Θ = µθ
(2.7)
(2.8)
These are the three momentum equations (2.1 – 2.3) for the zonal velocity u, the
meridional velocity v and the vertical velocity w; the thermodynamic equation
(2.4) for the potential temperature θ; the mass conservation equation (2.5) and
the equation for the non-conserved geopotential (2.6). Density can be diagnosed
from:
∂η φ = −αµ.
(2.9)
and pressure from the equation of state:
p = p0 (Rd θ/p0 α)γ ,
(2.10)
where γ = cp /cv = 1.4 is the ratio of the heat capacities and Rd is the gas constant
for dry air.
Moisture is included into (2.1–2.6) by introducing mass mixing ratios and
conservation equations for hydro meteors rather than introducing source terms
in the mass conservation equation (2.5). The coupling to the dry air mass µd is
retained. The moist Euler equations become:
11
∂t U + (∇ · Vu)η + µd α∂x p + (α/αd )∂η p∂x φ = FU
(2.11)
∂t V + (∇ · Vv)η + µd α∂y p + (α/αd )∂η p∂y φ = FV
(2.12)
∂t W + (∇ · Vw)η − g [(α/αd )∂η p − µd ] = FW
(2.13)
∂t Θ + (∇ · Vθ)η = FΘ
(2.14)
∂t µd + (∇ · V)η = 0
h
i
∂t φ + µ−1
(V
·
∇φ)
−
gW
=0
d
η
(2.15)
(2.16)
∂t Qv + (∇ · Vqv )η = FQv
(2.17)
∂t Qc + (∇ · Vqc )η = FQc
(2.18)
∂t Qr + (∇ · Vqr )η = FQr
(2.19)
∂t Qi + (∇ · Vqi )η = FQi .
(2.20)
Qv,c,r,i are the mass-coupled mixing ratios for water vapor, cloud, rain and ice.
The moist potential temperature is approximated by θm = θ(1 + 1.61qv ) and the
diagnostic relations for the full pressure and the inverse density are:
p = p0 (Rd θm /p0 αd )γ ,
∂η φ = −αd µd .
(2.21)
(2.22)
The full parcel density α−1 is calculated as the sum over all mixing ratios times
the dry density:
α = αd (1 + qv + qc + qr + qi )−1
(2.23)
where qη are the uncoupled mixing ratios.
Removing the hydrostatically balanced part of the pressure gradient yields
a set of equations which are spatially discretized on a Arakawa C-grid. The temporal discretization separates low-frequency from high-frequency acoustic modes.
The low-frequency modes are integrated with a third-order Runge-Kutta (RK3)
12
scheme.
The efficiency of the solver is increased because the high-frequency
modes are calculated on shorter time steps at each RK3 integration while the
low-frequency modes are held constant. The acoustic time step and the choice
of the accuracy of the flux divergence (2nd to 6th order) determine the stable
Courant number. In this study we used 5th order accuracy and the theoretical
Courant number is Crtheory = 1.42 (Skamarock et al. 2005). This large number
comes at the expense of 3 calculations (plus acoustic mode) per time step. For a
three-dimensional experiment, the time step should satisfy:
∆tmax <
Crtheory ∆x
√
·
umax
3
(2.24)
The maximum time step with ∆x = 8333m and umax = 100ms−1 is approximately 70 s. The acoustic step however uses a forward-backward time integration with a different Courant number. The maximum allowed time step is
√
Crmax = cs ∆τ /∆x < 1/ 2 but a more conservative estimate for the acoustic step
is:
∆τ <
∆x
.
2 · cs
(2.25)
With the sound speed cs = 300ms−1 and ∆x = 8333m, the acoustic time step is
approximately 14 s. For the WRF simulations we have specified a time step of
60 s and a 1:6 ratio of RK3 time step to the acoustic time step. The acoustic
time step is therefore 10 s and satisfies the conservative estimate of the Courant
number.
To resolve the boundary layer accurately we use 69 vertical levels on a
stretched vertical grid with 35 levels below 1500 m. (The model sigma levels are
listed in APPENDIX A). The model is either run in a 2 or 3 domain configuration.
The 2-domain (1 nest) configuration is used for the validation of the model while
13
the 3-domain (2 nests) is used to achieve higher horizontal resolution. Time step
and grid size ratios are chosen to satisfy the Courant number constraints shown
above.
2.1.1. Special consideration of turbulence and entrainment
The boundary layer scheme and the WRF-model itself are relatively new
developments and have not been used very widely. The GB01 has performed well
in the tropics and subtropics (McCaa and Bretherton 2004) but it has not been
tested in combination with the WRF-model. The GB01 offers some advantages
over other boundary layer schemes. It attempts to simulate the tight coupling
between cloud formation, convective turbulence, radiative and surface fluxes and
it can handle the internal stratification of a cloud topped boundary layer (Grenier
and Bretherton 2001), meaning that it is able to resolve several convective layers
within the ABL.
This analysis focuses on the boundary layer and it is therefore worthwhile
to examine its numerical representation. The boundary layer model is based
on Grenier and Bretherton 2001 and uses a 1.5 order turbulence closure. The
original numerical implementation was provided by McCaa (McCaa 2001, personal
communication). Turbulent flux divergence, namely
∂
hu0 w0 i
∂z
and
∂
hv 0 w0 i,
∂z
is
discretized vertically by:
0
hw0 Xh,m
i = Kh,m ρg
δX
δp
(2.26)
where δX is a generalized vertical difference operator. The eddy diffusivities of
Kh for conserved thermodynamic variables and the eddy viscosity Km are related
to the turbulent kinetic energy e, a master length scale l and stability functions
Sh,m :
14
√
Kh,m = l eSh,m
(2.27)
The length scale l is based on the concept that in a turbulent environment a vertically displaced air parcel will transport its perturbation velocity u0 a characteristic
distance l and then creates a fluctuation in the turbulence. The specification of
the stability function Sh,m follows the definition of Galperin et al.(1988).
In the case of convective and stable ABLs, the master length scale is specified after Blackader (1962):
l = kz/(1 + kz/λ)
(2.28)
with the asymptotic length scale λ = ηl zi , ηl = 0.1. In the case of a decoupled ABL
the parcel length scale lp is more appropriate. The parcel length scale lp consists
of an upward and downward length scale lu and ld , and is calculated based on
the buoyancy of the parcel at the point where it would reach zero velocity. The
upward and downward length scales are limited by the distances from the parcel
to the ABL top and the surface, respectively.
lp = ηl (lu + ld )
(2.29)
The parcel length scale is incorporated into the definition of the master length
scale:
l = kz/(1 + kz/lp )
(2.30)
The relationship to the Blackadar length scale is obvious for a single convective
layer when the parcel length scale lp is identically lp = ηl zi = λ.
The model is closed by specifying an entrainment velocity we at the base of
the ABL inversion (subscript i). The Turner-Deardorff closure approach (Turner
15
1973) relates we to an eddy length scale L and a velocity scale U , the entrainment
efficiency A and the buoyancy jump ∆i b across the inversion :
we = AU 3 /(L∆i b)
(2.31)
With the local entrainment closure approach (GB01), the eddy size and velocity
√
scales are taken from just below the inversion and set to L = li and U = ei .
Thus we becomes
√
ei ei
we = A
li ∆i b
(2.32)
In (2.32) the entrainment efficiency A is chosen following GB01. The convective
ABL is assumed to be topped with an infinitely thin inversion (GB01) at some
grid point i at the inversion height. The turbulent fluxes of X at this height are
parameterized following (Lilly 1968):
hw0 X 0 ii = −we ∆i X
(2.33)
The ABL model uses the the so-called ’restricted inversion’ (GB01) method to
calculate the inversion jump ∆i X of variable X across the inversion. In this
method, the inversion is restricted to lie on a flux level of the model. The layer
which contains the inversion is therefore composed of a mix of ABL properties
from below the inversion and free atmospheric properties from above. Grenier
and Bretherton (GB01) refer to this layer as the ’ambiguous layer’. This method
is computationally simple and allows deepening of the boundary layer implicitly through entrainment. Entrainment acts to reduce the difference between the
ambiguous layer and the well mixed ABL properties, thereby reducing the stratification until ultimately the inversion is diagnosed to lie one grid level above the
previous inversion.
16
2.2. Comparison of simulated and observed wind response
The WRF-model and the GB01 parameterization are relatively new and
we feel that it is necessary to quantify the performance of the WRF-model/GB01
combination. Therefore, the new WRF-model-GB01 version of the model is tested
in a realistic simulation over the Agulhas return current. South-east of the Republic of South Africa, the Agulhas return current meanders with quasi-stationary
troughs and crests that separate the cold Southern Ocean from the warmer Indian Ocean (O’Neill et al. 2005). The accompanying SST distribution contains
sharp mesoscale SST gradients (Fig. 2.1) and the large-scale SST gradient is to
the north, perpendicular to the mean zonal wind. The land influence is minimal.
In combination with the availability of satellite observations of both SST and
wind speed, the Agulhas region is an ideal environment for model validation and
in particular to test the performance of the boundary layer scheme over the open
ocean.
The SST data is obtained from the Advanced Microwave Scanning Radiometer (AMSR) onboard the Earth Observing System (EOS)-Aqua system
(Chelton and Wentz 2005). The AMSR measures polarized brightness temperatures at six microwave frequencies. The resolution of the derived SST product is
about 58 km and all available locations for the month of July 2002 were averaged
onto a 0.25◦ grid. The rms accuracy for a single observation is on the order of
0.5◦ C but the random error is much smaller for the 30 day average considered in
this study.
The initial fields and boundary condition for the 30 day WRF-model simulation is obtained from NCEP’s Global Data Assimilation System (GDAS). The
Global Final Analyses data sets consist of simulations over the entire globe on a
17
1.0◦ ×1.0◦ grid of the spectral Medium Range Forecast model. Spectral modes are
interpolated on pressure levels and are available every 6 hours.
The period of July 2002 was motivated by the observed annual cycle of
coupling coefficients (O’Neill et al. 2005) with largest coupling coefficients during
austral winter. The bottom boundary consists of averaged AMSR SST and is
held constant throughout the simulation. The analyzed results from the WRF
simulation are averaged over the last 28 days to allow spin-up time of 2 days at
the beginning of the simulation.
The WRF-model near-surface wind that is used in the comparison is the
10m wind product from the model. The model 10m wind product and the wind
speed of the lowest model level are virtually identical because the lowest model
level lies on average at 12m. The QuikSCAT wind product is a 10m equivalent
neutral wind derived from surface roughness. Over the worlds ocean the nearsurface stratification is usually slightly unstable and on average 10m anemometer
measurement are about 0.2ms−1 lower than scatterometer 10m estimates (Mears
et al. 2001). The accuracy for QuikSCAT wind vectors has been estimated to be
about 1.7ms−1 in wind speed and about 14◦ in direction (Chelton and Freilich
2005).
The agreement between the WRF-model simulation and the satellite observation of 10 m wind is remarkable. Maps of high-pass filtered wind speed and
high-pass filtered SST (Fig. 2.2) emphasize the very good agreement for the purpose of this dissertation. Higher wind speed is simulated over relatively warm SST
and lower wind speed over relatively cool SST. The model zonal wind (Fig. 2.3)
in the eastern part of the domain is slightly smaller than the observed QuikSCAT
zonal wind. The observed and modeled zonal wind speed in the western part of
the domain agree fairly well. The relationship between the meridional component
18
and SST can also be seen in Fig. 2.4 but is less obvious. This is in part due to the
generally lower meridional wind speed magnitudes which reduces the magnitude
of the wind response. The zonal and meridional components of the WRF-model
show generally less high wavenumber details than the QuikSCAT observations,
which indicates that the effective resolution of the WRF-model is less than the
QuikSCAT observation, potentially because the SST forcing had to be interpolated onto WRF-model grid and is therefore smoother than the SST that underlies
the QuikSCAT observations. On the other hand, there is no guarantee that the
high wavenumber details in QuikSCAT are reliable. The 30◦ longitude by 10◦ latitude spatial high-pass 2-D loess-filter (Cleveland and Devlin 1988; Schlax et al.
2001) removes the slow varying background field and therefore, large scale biases
between model and observations are not obvious in the comparison.
The relation between surface winds and SST can be quantified in terms of
wind divergence and curl as functions of downwind and crosswind SST gradients,
respectively (Chelton et al. 2001; 2004; O’Neill et al. 2003; 2005). The prime
symbol in Eqn. 2.34 and Eqn. 2.35 denote the high-pass filtered part of the variable
and is the same 30◦ longitude by 10◦ latitude loess-filter used by O’Neill (2003
and discussion therein).
∇ · u0 o = s1 · (∇T · ûo )0
∇ × u0 o = −s2 · (∇T × ûo )0
(2.34)
(2.35)
The derivative wind fields of QuikSCAT were computed using in-swath results
which are averaged over the month of July 2002. Note that O’Neill (2005) used
surface wind stress to compute the coefficients while here surface wind speed is
used, which effects the magnitude of the coefficients. The methods used here are
otherwise identical. The derivative fields of WRF-model were computed from daily
19
vector averages of the wind and then averaged for the month of July 2002. Each
coefficient s1,2 characterizes the wind-SST relation for the entire region of interest
and can be used to compare WRF simulations and QuikSCAT observations in a
quantitative way. For this comparison, the divergence and curl as well as the down
and crosswind SST gradients have been calculated by Larry O’Neill (Oregon State
University) to ensure consistency with previous studies. The binned scatterplots of
divergence and curl as functions of SST gradient are shown in Fig. 2.5 and Fig. 2.6,
respectively. The very good agreement found for the zonal and meridional wind
components (Fig. 2.3 and Fig. 2.4) is quantified by the similarity of the coefficient
s1 for WRF-model and QuikSCAT surface wind divergence (Fig. 2.5) as well as the
agreement for the surface wind curl coefficient s2 (Fig. 2.6). The difference between
the coefficient for the divergence s1 and the curl s2 has been observed by Chelton
et al. (2001; 2004) and O’Neill et al. (2003; 2005). One potential mechanism for
different coupling coefficients may be the difference in vertical shear between the
zonal and meridional wind component and we will discuss this mechanism further
in Chapter 4 and 5. The current chapter, however, only focuses on the comparison
between the WRF-model simulation and satellite observations.
The perturbation crosswind and downwind components of the SST gradient
may be expressed similarly as:
∇T · u0 o = |∇T| · sin θ0
(2.36)
∇T × u0 o = |∇T| · cos θ0
(2.37)
where the counterclockwise angle θ0 is defined as:
0
0
−1 ∇T × u o
θ = tan
∇T · u0 o
(2.38)
(Chelton et al. 2001; O’Neill et al. 2005).
These relations provide an additional method for quantifying the coupling between
20
surface wind response and SST gradient. The binned scatterplots of divergence
and curl as function of counterclockwise angle θ0 are shown in Fig. 2.7 and Fig. 2.8,
respectively. The simulated curl and the observed curl agree very well. However
the simulated response of the divergence is almost two times larger than the
observed. This quantitative discrepancy is surprising given the otherwise good
quantitative agreement between simulation and observation.
2.2.1. Conclusion
The WRF-model successfully reproduces the QuikSCAT satellite observations of the wind response to observed mesoscale SST anomalies. The location of
local wind maxima relative to local SST maxima is in very good agreement with
the observations. The quantitative relationship between the wind and SST perturbations is expressed in terms of downwind and crosswind divergence and curl
and is used to summarize the performance of the WRF-model. The agreement
with the QuikSCAT wind divergence and curl as functions of down and crosswind
SST gradients is remarkable. The largest quantitative difference is found between
for divergence as a function of counterclockwise angle θ0 , while a good agreement
is found for the curl as function of θ0 . The overall good agreement provides confidence for using the WRF-model to investigate the atmospheric boundary layer
response to midlatitude SST variability. The bias between the surface wind estimate from QuikSCAT and the model surface wind might be due in part to the
differences in stability and resolution between the neutral 10m wind estimates
from QuikSCAT and the simulated 10m winds from the WRF-model. On average, the marine boundary layer is slightly unstable and therefore 10m anemometer
measurement are about 0.2ms−1 lower than scatterometer 10m neutral estimates
21
(Mears et al. 2001), which is in agreement with the observed differences between
WRF-model and QuikSCAT in this comparison study (Fig. 2.3). The WRF-model
contains less high wavenumber details than the QuikSCAT observations which indicates that the effective resolution of the WRF-model is less than the QuikSCAT
observations. The mean difference does not effect the coupling coefficients since
they are calculated based on perturbation winds alone.
22
FIGURE 2.1. Map of averaged July 2002 AMSR SST. Shaded SST contours
are shown in 2◦ C intervals. Gray rectangles denote the boundaries of the WRF
simulation. Missing SST data around islands are masked white and interpolated
for the aqua-planet simulation. SST contours are used as overlays in subsequent
map figures.
23
FIGURE 2.2. Comparison between WRF simulation (top) and QuikSCAT observation (bottom) averaged over the month of July 2002. Maps show inner WRF
domain simulations and corresponding QuikSCAT observations of wind speed perturbations. Contours of SST perturbation are overlaid. The contour interval is
1◦ C.
24
FIGURE 2.3. As in Fig. 2.2, except for zonal wind. SST contour interval is 2◦ C.
25
FIGURE 2.4. As in Fig. 2.2, except for meridional wind. SST contour interval is
2◦ C.
26
WRF Perturbation Divergence
2
1.5
m/s per 100km
1
0.5
0
−0.5
−1
−1.5
Slope=0.55
−2
−2
−1.5
−1
−0.5
0
0.5
1
Perturbation Downwind SST Gradient (°C per 100km)
1.5
2
QuikSCAT Perturbation Divergence
2 WRF Perturbation Curl
2
1.5
m/sm/s
per per
100km
100km
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2
−2
−2
Slope=0.59
−1.5
−1.5
−1
−0.5
0
0.5
1
Perturbation
Downwind
SST
Gradient
(°C
per
100km)
−1
−0.5
0
0.5
1
Perturbation Crosswind SST Gradient (°C per 100km)
Slope=−0.39
1.5
1.5
2
2
m/s per 100km
FIGURE QuikSCAT
2.5. Binned
scatterplots
of the relationships between the perturbaPerturbation
Curl
tion divergence
and the perturbation downwind SST gradient: (top) perturbation
2
divergence of WRF-model and (bottom) perturbation divergence of QuikSCAT
1.5
observation.
The points are the means within each bin computed from 28 consecutive days, and the error bars are ±1 std dev of the means within each bin. The
1
lines through the points represent least square fits of the binned overall means to
straight0.5
lines.
0
−0.5
−1
−1.5
−0.5
−1
−1.5
Slope=0.55
−2
−2
2
2
−1.5
−1
−0.5
0
0.5
1
Perturbation Downwind SST Gradient (°C per 100km)
1.5
2
27
QuikSCAT Perturbation Divergence
WRF Perturbation Curl
1.5
1.5
m/s
per
100km
m/s
per
100km
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−2−2
−2
Slope=0.59
Slope=−0.39
−1.5
−1.5
−1
−0.5
0
0.5
1
−1
−0.5
0 Gradient
0.5(°C per 100km)
1
Perturbation
Downwind SST
Perturbation Crosswind SST Gradient (°C per 100km)
1.5
1.5
2
2
QuikSCAT Perturbation Curl
2
1.5
m/s per 100km
1
0.5
0
−0.5
−1
−1.5
Slope=−0.39
−2
−2
−1.5
−1
−0.5
0
0.5
1
Perturbation Crosswind SST Gradient (°C per 100km)
1.5
2
FIGURE 2.6. As in Fig. 2.5, except for the perturbation curl and the perturbation
crosswind SST gradient.
28
WRF Perturbation Divergence
2
1.5
m/s per 100km
1
0.5
0
−0.5
−1
−1.5
Amplitude=0.91
−2
−180
−135
−90
−45
0
θ′ (deg)
45
90
135
180
QuikSCAT Perturbation Divergence
2 WRF Perturbation Curl
2
1.5
m/sm/s
per per
100km
100km
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−180
−2
−180
Amplitude=0.55
−135
−90
−45
−135
−90
−45
0
θ′ (deg)
0
θ′ (deg)
45
Amplitude=0.55
90
135
180
45
90
180
135
m/s per 100km
FIGURE 2.7.
Binned
scatterplots
of the angular relationships between the wind
QuikSCAT
Perturbation
Curl
and the 2SST field. Top panel shows WRF perturbation wind divergence as a
function of the angle between crosswind and downwind SST gradient. Bottom
1.5
panel shows
QuikSCAT perturbation wind divergence as function of the angle
between crosswind and downwind SST gradient. The points are the means within
1
each bin computed from 28 consecutive days, and the error bars are ±1 std dev
of the means
within each bin. The lines through the points represent least square
0.5
fits of the binned overall means to a cosine curve.
0
−0.5
−1
−1.5
−1
−1.5
Amplitude=0.91
−2
−180
2
2
−135
−90
−45
0
θ′ (deg)
45
90
135
45
45
Amplitude=0.55
Amplitude=0.55
90
135
90
135
180
29
QuikSCAT Perturbation Divergence
WRF Perturbation Curl
1.5
1.5
m/s
per
100km
m/s
per
100km
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
−1.5
−1.5
−2
−180
−2
−180
−135
−135
−90
−90
−45
−45
0
0
θ′ (deg)
θ′ (deg)
180
180
QuikSCAT Perturbation Curl
2
1.5
m/s per 100km
1
0.5
0
−0.5
−1
−1.5
Amplitude=0.42
−2
−180
−135
−90
−45
0
θ′ (deg)
45
90
135
180
FIGURE 2.8. As in Fig. 2.7, except for the WRF perturbation curl (top) and
the QuikSCATperturbation curl as a function of the angle between crosswind and
downwind SST gradient. The lines through the points represent least square fits
of the binned overall means to sine.
30
3. ABL RESPONSE TO MIDLATITUDE SST FRONTS
In this chapter we perform a set of numerical experiments to examine the
response of the atmospheric boundary layer (ABL) to mid-latitude sea surface
temperature (SST) fronts. Recent observational studies have established that
small scale (≈ 10–1000 km) SST variations are positively correlated with surface wind variations. The correlation has been documented for the northeast
Pacific equatorial front through in situ measurements (e.g., Wallace et al. 1989,
Hashizume et al. 2002). Analyses of satellite observations in numerous regions
of the world’s ocean (see review by Chelton et al. 2004) have documented that
the near-surface divergence is a linear function of the SST gradient and that the
coupling coefficients are surprisingly constant over the world’s ocean where strong
SST gradients persist.
Several numerical experiments have been performed to interpret the observations (de Szoeke 2003, Small et al. 2003, Samelson et al. 2006) with somewhat
contradictory results. Depending on the scale of the SST anomalies, every term
of the momentum budget that represents an important process in the surface
wind–SST coupling has been used to explain the mechanism. Mahrt (1972) noted
that for southerly flow across the Equator with non-zero zonal component, the
zonal wind component just north of the Equator is opposite to the northern hemisphere Ekman balance and thus accelerates the flow. On time scales of one month
and longer, Lindzen and Nigam (1987) suggested that hydrostatic pressure adjustments with positive pressure perturbations over cold water and negative pressure perturbations over warm water generate near-surface pressure gradients that
would accelerate the flow. Near the Equator such a pressure gradient would generate strongest winds over the SST front in contrast to the observations. Small et al.
31
(2005) overcomes this discrepancy by arguing that the slowly adjusting air temperature shifts the pressure adjustment downstream, therefore generating largest
wind speeds over the warmest water and vice versa. Yet another group of explanations emphasizes the importance of vertical divergence of turbulent momentum
flux. Sweet et al. (1981) and Wallace et al. (1989) argued that a stability-induced
increase in the vertical turbulent mixing of horizontal momentum downward increases the near surface velocity over warm water. Over cool water, increased
stability reduces vertical mixing and the surface velocity decreases while the flow
aloft accelerates. This mechanism was also used by de Szoeke and Bretherton
(2004) to explain the results of their LES simulation. An observational study
by Mahrt et al. (2003) over the Gulf stream finds evidence for stability-induced
vertical momentum mixing. On small scales of less than 10 km, they found that
strong southerly momentum is mixed down surface below the 33 m flight level
and plays a role in rotating the flow over the the warm side of the Gulf stream.
Samelson et al. (2006), on the other hand, noted that the boundary layer
average flux divergence (τs −τh )/H is constant for steady conditions with constant
pressure gradient forcing and no entrainment stress at the top of the boundary
layer. Under these conditions the surface stress and hence the surface wind speed
changes with changes in the depth of the boundary layer H. Samelson et al. (2006)
assumed no entrainment stress for simplicity, but Stevens et al. (2002) and others
have found that entrainment mixing of ABL air with the free atmosphere is an
important process in the time averaged ABL momentum budget.
The complexity of surface wind adjustments in the vicinity of real world
SST variations, in combination with undersampling of the ABL vertical structure
by satellite and in situ instruments, make it difficult to isolate the governing physical processes. Numerical simulations of the ABL flow over idealized SST fronts
32
provide the possibility of identifying the governing ABL adjustment mechanisms
with the objective of interpreting ABL response in more realistic conditions. To
be useful, a simulation must simultaneously resolve the complex vertical structure
of the ABL adjustment in the vicinity of the front while covering a spatial domain
large enough to simulate the important physical processes already mentioned such
as pressure adjustments and zonal advection. De Szoeke and Bretherton (2004)
took a Lagrangian approach using an eddy resolving large eddy simulation (LES)
column model with prescribed large-scale time-varying conditions of underlying
SST, large-scale pressure gradients and overlying free atmosphere temperature and
humidity to advect the 3 km cubic domain across the Equator near 95◦ W. They
found that the vertical wind structure and surface wind in their model agrees well
with the vertical mixing effect of Sweet et al. (1981) and Wallace et al. (1989).
They also noted that, due to the model design, differential advection at different
heights is not simulated. Moreover, because the horizontal pressure gradient force
driving the LES model is prescribed, de Szoeke and Bretherton (2004) are unable
to determine the role of meso- and synoptic scale pressure adjustments.
Small et al. (2003, 2005) used a finite difference numerical model to simulate a large domain in the eastern tropical Pacific of roughly 30◦ latitude ×30◦
longitude, but with limited vertical resolution of 12 levels below 800 hPa. They
conclude that a lagged pressure adjustment can explain surface wind acceleration
in the vicinity of the equatorial SST front but note that modeled acceleration
of surface wind near the front is too small. Although they did not find any evidence for the Wallace et al. mechanism in their simulations, they note that aircraft
observations during the EPIC field program (Raymond 2004) show anomalous turbulent momentum flux convergence that could increase surface wind acceleration
to the observed levels.
33
Another LES-type simulation is presented in Samelson et al. (2006) for the
flow of a shallow boundary layer across an SST front. They conclude under the
condition of a constant pressure gradient across the SST front that the change in
vertical momentum mixing cannot accelerate surface wind and that a deepening
of the boundary layer is responsible for the increased surface stress.
Because of the various model designs and setups, previous numerical simulations are unable to depict all essential processes simultaneously. These studies
tend to favor one mechanism while not being able to exclude others. This study
overcomes these deficiencies by adopting a mesoscale numerical model with high
vertical resolution. This simulation uses a total of 69 levels on a stretched vertical
grid. The lowest model layer is at 22 m with more than 30 levels in the lowest
1500 m. The model allows for simulation over a large-scale domain and also resolves differential horizontal advection of momentum and heat in shallow layers
near the surface.
34
3.1. Simulations with idealized SST fronts
This section describes two basic model simulations which use simplified
bottom boundary conditions to study the marine boundary layer response to
mid-latitude sea surface temperature (SST) fronts. The simulations are performed
using the WRF-model with the Grenier-Bretherton ABL scheme (GB01, detailed
discussion can be found in Section 2.1.1). The lateral boundary conditions of the
WRF-model outer domain along 42◦ E and 88◦ E and between 33◦ S and 57◦ S
are prescribed by an aqua-planet version of the global circulation Community
Atmosphere Model (CAM) (Fig. 3.1). The CAM simulations were performed
by Eric Maloney (Oregon State University) with forcing by steady, latitudinally
varying SST over the hypothetical ocean-covered planet and time-invariant solar
radiation. The symmetric SST pattern eliminates the influence of the land mass
distribution and generates nearly symmetric time-averaged meteorological fields
about the meteorological equator. We chose an arbitrary 30 day time period after
the three year spin up to initialize and to calculate time-dependent lateral boundary conditions for the WRF-model simulations. At the initial time of the WRF
simulation the SST distribution was modified to include an SST-front extending
in the north-south direction (Fig. 3.1). Two simulations were performed. The
SST-front consists either of a positive 4 K change or a negative 4 K change over a
distance of approximately 0.5◦ longitude. The sign of the SST change across the
front is the only difference between the two simulations. Therefore, any difference
in the response between the two simulations must be directly related to the SST
change. Experiments with east-west SST-fronts were also considered but abandoned for practical reasons. Our strategy of analyzing the anomalies associated
with the SST-front could not be applied to the east-west cases. It is difficult to
35
partition the time-averaged response to a mesoscale south-to-north SST gradient into its large-scale and mesoscale components. Large scale effects would most
likely dominate the small scale adjustments and the small frontal anomalies would
be absorbed into the adjustment of the mean background flow.
The use of a high vertical resolution numerical weather prediction model
overcomes limitations of the previous numerical studies. For practical reasons
Large Eddy Simulation (LES) modelers have found it necessary to prescribe pressure distributions and to place the upper boundary below the tropopause. These
practical limitations artificially suppress the response of the pressure field to differential heating and prevent dynamical adjustments of the pressure and wind
fields. Regional Climate Models (RCM) and Global Circulation Models (GCM)
allow for differential heating and pressure adjustment. However, the finite difference numerical models that have been used thus far have been limited by the
vertical resolution which is crucial for studies of boundary layer processes.
This WRF-model simulation overcomes some of the limitations of previous
regional finite difference numerical simulations by increasing the vertical resolution in the boundary layer in combination with the use of a state-of-the-art
boundary layer scheme. The boundary layer scheme is a layer-orientated vertical turbulent mixing scheme that allows for multiple vertically stacked turbulent
layers (Bretherton et al. 2004). Changes in SST are communicated throughout
the atmosphere in the inner domains of the model. Far away from the region of
interest at the boundaries of the outermost domain, the model is constrained to
match the results from global aqua-planet simulations. This provides the ideal
test environment for simulations of various midlatitude SST distributions.
We believe that the ABL adjustment mechanism depends on the scale
of the SST anomaly. Previous numerical simulations (e.g., Small et al. 2003,
36
Hafner and Xie 2003, Small et al. 2005) have relied on observed SST distribution
that contain a broad range of SST gradients on scales from 50 km to 1000 km
which potentially underestimate sharp SST gradients similar to the ones observed
by Mahrt et al. 2004 and de Szoeke and Bretherton (2005). Consequently, the
previous regional model results might not contain the small scale ABL response
to sharp SST gradients. From our numerical simulations of step-like SST-fronts,
we extend the wavenumber cut-off to larger values. We hope to simulate the small
scale response of the ABL to sharp SST gradients and to predict the magnitude
of the pressure gradient changes associated with the SST change as well as the
phase relation of the pressure anomaly relative to the SST-front.
The initial fields and boundary conditions were provided by Eric Maloney
from an integration of the Community Atmosphere Model version 2 (CAM2). The
simulation represents the general circulation over an ocean-covered planet with
meridionally symmetric SST and radiation. The CAM2 SST boundary condition
is time invariant and the latitudinal structure is represented by a smoothed cosinefit of observed zonally-averaged SST from the Advanced Mircrowave Scanning
Radiometer (AMSR). The SST for the regional WRF simulation is derived from
the aqua-planet simulation by altering only the lower boundary condition of the
SST distribution. In the first experiment (hereafter denoted by P4 or “plus 4K”)
the background SST is increased by 4 K from approximately the middle of the high
resolution inner-most domain to simulate a step increase in SST in the east-west
direction, and creating, thereby, a north-south SST front through the center of the
domain (Fig. 3.1). Similarly, for the second experiment (hereafter denoted by M4
or “minus 4K”), the background SST is decreased by 4 K to simulate a negative
north-south SST front. The orientation of the SST front is almost perpendicular to
the prevailing west-northwest background wind and the two experiments therefore
37
simulate cold-to-warm (P4) and warm-to-cold (M4) transitions. The outer WRF
domains cannot resolve the sharpness of the SST front, creating a mismatch at
the boundaries. In addition, there is an SST front created at the outer boundary
of the WRF domain where the perturbed SST values are blended with those used
in the CAM2 simulation. However, the boundaries of the outer domain are far
away from the inner domain’s SST front and most of the inconsistency is absorbed
by a numerical “sponge layer” at the edge of the outer domain.
We will see that both simulations M4 and P4 show large responses in the
atmospheric boundary layer. The response is not linear, and this asymmetry
between the cold-to-warm response and warm-to-cold response indicates that adjustment mechanisms may differ in the M4 and P4 simulations. The results of
each experiment are described in the following two sections and are compared in
the section thereafter.
We will show that the region of the largest response occurs in a relatively
narrow transition region over and just downstream from the largest SST gradients.
In this region, the flow is adjusting rapidly towards the balanced state found in the
far-field, away from the region of strong SST gradient. The flow in the far-fields is
in an Ekman balance between pressure gradient force, Coriolis force and turbulent
stress convergence. The flow across the SST gradients changes the depth of the
boundary layer. A deeper boundary layer and stronger wind speeds are found over
the warmer water and a shallow boundary layer and weaker wind speeds are found
over the cold water in the far-fields (compare with Samelson et al. 2006). In the
narrow SST region, the ABL adjusts to the rapidly changing conditions. There are
however qualitative and quantitative differences in the adjustment mechanisms of
P4 and M4 in the transition zone.
38
3.1.1. Methodology
We use 30-day simulations of the regional mesoscale WRF-model to examine the ABL responses to positive and negative step changes in SST. The lateral
boundary conditions for the WRF-model are taken from a single CAM2 simulation
over a symmetric aqua-planet after a 3-year spin-up period.
We now introduce some terminology that will facilitate the discussion of
the P4 (Section 3.2) and the M4 (Section 3.3) experiments. The change in SST
occurs over a distance of approximately 0.5◦ longitude and remains constant thereafter. The area between 60.5◦ longitude and 61.◦ longitude will in the following
be referred to as the SST front or the frontal region. The width of the frontal
region was limited by the resolution of the model’s finite difference scheme. It
was our intent to make the frontal region as narrow as possible to study the
ABL response to a “step” change in SST. Several experiments were performed
to estimate an ’effective resolution’ of the model. The horizontal finite difference
scheme of the WRF-model produces a computationally stable ‘2∆x’ oscillation
due to spatial undersampling as the front width approaches the grid spacing. The
width of the SST front was therefore chosen to be as narrow as possible while
suppressing the computational mode. The functional form of the step change in
SST is given as a hyperbolic cosine function. The sharp SST front can be well
resolved with by a minimum of 6 grid points in longitudinal direction. The wind
direction is consistently from the west–northwest, and therefore the wind direction
is almost perpendicular to the SST front. Short periods of easterly surface winds
due to synoptic variability were found during the 30 day run. The flow of boundary layer air during these periods opposed the prevailing direction relative to the
SST-front. Although including these periods did not alter the time-averaged mo-
39
mentum budget results qualitatively, the inclusion complicated the analysis and
obfuscated the results. As will be discussed in section 3.4 where the P4 and the
M4 are compared, physically different mechanisms are present for cold-to-warm
and warm-to-cold SST fronts. The simplest and most effective way to avoid the
complication is to exclude the periods of reversed wind direction from the analysis.
From the meridionally averaged longitude-height cross-sections for the P4
experiment (Figs. 3.7 – 3.12) and from the M4 experiment (Figs. 3.19 – 3.24) it becomes apparent that most of the adjustment of the ABL takes place in a relatively
narrow transition zone about 100 km wide beginning near the upstream frontal
boundary and extending slightly further than the downstream frontal boundary.
The magnitude of the zonal momentum advection term (Figs. 3.5, 3.18) can be
used to characterize the regions of strong and weak adjustment and to separate
the model domain into three regions (Fig. 3.6). Upstream and away from the SST
front, the momentum advection is approximately zero and longitudinal changes
in meteorological variables are small. This region is in Ekman balance and therefore is referred to as the “equilibrium region”. Near and just downstream from
the SST front, advection is a dominant term in the momentum balance and the
atmosphere is adjusting rapidly to the increase of SST. This region is hereafter
called the “transition region” or “advection region”. Advection becomes negligible
again downstream of the transition region. However, the atmosphere is still not in
balance and continuing adjustments are taking place, although at a much slower
rate compared to the transition region. This third region found downstream of
the transition region is called the “quasi-equilibrium” region. The region is nearly
in Ekman balance but thermodynamically driven pressure adjustments are still
taking place. Each of these regions will be separately discussed for the P4 and
M4 experiment in sections 3.2 and 3.3, respectively.
40
An important characteristic of the boundary layer is its depth H. Traditionally, the boundary layer depth is thought to be the height above the ground
where the direct influence from the surface vanishes. The GB01 ABL-scheme calculates a boundary layer depth based on the buoyancy frequency of the layer. The
parameterization identifies the top of the convective layer as the location where
(N 2 l2 )layer ≥ −0.5(N 2 l2 )avg , where N is the buoyancy frequency, l is the turbulent length scale, and the subscript “avg” indicates an average over all convective
layers below a given layer indicated by subscript “layer” (Grenier et al. 2001).
This ABL scheme predicts very shallow boundary layers in stable conditions. In
our analysis, however, we are interested in the depth of the momentum boundary
layer, which can be significantly deeper in stable conditions than the buoyancy
frequency definition used by GB01 would suggest. Therefore, the GB01 definition
of H cannot successfully be used as the momentum boundary layer depth in all
situations. An alternative definition of H as the level of zero stress is also deficient
because we wish to analyze the averaged as opposed to instantaneous momentum
budgets. There is a difference between 1) averaging the stress first and then calculating H and 2) calculating H for each observation and subsequently averaging
the individual heights. To illustrate this difference, assume H to be the level
where stress becomes negligible and imagine two instances, one where the stress
vanishes at 500 m and one at 1000 m. Averaging the individual heights results in
an average boundary layer depth of 750 m. However, the averaged stress will have
significant values above 750 m (in this simple case exactly 1/2 of the 1000 m deep
boundary). Estimating H from the averaged stress would therefore result in a
boundary layer deeper than 750 m. Large variability of the boundary layer depth
can increase the discrepancies. The simple example also shows that in order to
obtain a height closer to 750 m from averaged stress profiles, one needs to choose
41
larger-than-zero values of the stress to define a consistent averaged boundary layer
depth.
Instead of identifying H by defining a stress threshold level, we use a threshold value for stress convergence SH to find the depth of the averaged boundary
layer. To obtain a meaningful value of SH from the averaged momentum budgets,
we bin-average the instantaneous boundary layer depth from the ABL-scheme over
the upstream region (Figure 3.2) where the variability of turbulent stress convergence is smaller than anywhere else in the domain. The maximum frequency of
simulated ABL depth consistently occurs at a level of 450 m. The corresponding
value of turbulent flux convergence at H = 450 m (S450 ) is 2 × 10−4 ms−2 which
then may be used as an indicator of the momentum boundary layer depth across
the SST front. The average momentum boundary layer depth in the transition
and quasi-equilibrium region is therefore found by descending from the top of the
model column until a level where S = S450 is found. Descending from the top of
the model column avoids ambiguities where stress convergence changes sign within
the ABL. Turbulent stress convergence of 2 × 10−4 ms−2 corresponds roughly to
1/e of the surface value over our model domain.
Comparing this momentum boundary layer depth to the cross section of
potential temperature θ shows satisfying results. The momentum boundary layer
depth of the P4 run (Fig. 3.3) lies at the top of a well mixed layer where the vertical
gradient of potential temperature is small and just below a region of weakly stable
θ. From its upstream height of 450 m, the boundary layer grows rapidly to about
750 m within about 50 km. In the M4 run the averaged momentum boundary layer
depth can also be related to the potential temperature. The momentum boundary
layer depth follows the weakly stable potential temperature in the upstream region
and drops rapidly to about 200 m over the SST front and the ABL becomes very
42
stable. Subsequently, the boundary layer depth increases slowly along the contours
of θ.
In the P4 experiment, an internal boundary layer can be identified. This
internal layer is convectively driven by the surface buoyancy fluxes associated
with large positive sea-minus-air temperature difference. At the top of the internal boundary layer, air from the more stable upper portion of the ABL is
incorporated into the internal boundary layer through entrainment. The well
mixed thermodynamic properties of the internal layer can be used to estimate the
depth of the layer. We define the depth by the level where the vertical gradient
of θ changes sign from unstable (< 0) to stable (> 0) conditions. In the M4
run, no such internal boundary layer exists because of the stabilizing effect of the
decreasing SST downstream.
It is important to note that the thermodynamic boundary layer depth is
relatively constant across the SST-front. In this region of the ocean, the stratification of the lower atmosphere is slightly stable with a weak inversion at around
800 m (Fig. 3.3). The virtually constant thermodynamic boundary layer depth
lies above the momentum boundary layer. In large scale regions with strong convection, the momentum and thermodynamic boundary layer depth are expected
to be approximately equal. For the remainder of the chapter the concept of the
momentum boundary layer depth is more useful for our discussion. However, it is
important to be aware of the differences between the momentum and thermodynamic boundary layer depth to avoid confusion in Chapter 4 and 5.
43
3.1.2. Analysis of the simulated momentum budget
The simulations contain variability on time and space scales from days to
weeks. Our objective is the analysis of the average boundary adjustment that
persists on time scales of a month or longer. This lowpass component is obtained
by averaging over the last 28 out of the 30 days of simulation. The 2 days at
the beginning of the simulation are not used to exclude the period during which
the WRF-model is adding mesoscale detail to the CAM2 initial conditions and
adjusting to the imposed boundary conditions. The output interval is 4 hours
which gives 168 output periods spread over 28 consecutive simulation days. The
momentum budget for the zonal and meridional component can be written as
follows.
∂
∂u
∂u
∂u
1 ∂p
∂u
= −u
−v
−w
+ fv −
− hu0 w0 i
∂t
∂x
∂y
∂z
ρo ∂x ∂z
(3.1)
∂v
∂v
∂v
1 ∂p
∂
∂v
= −u
−v
−w
− fu −
− hv 0 w0 i
∂t
∂x
∂y
∂z
ρo ∂y ∂z
(3.2)
where the 28-day Reynolds average is denoted by an overbar. All terms in the
above equations have the usual meaning, and the terms
∂
hu0 w0 i
∂z
and
∂
hv 0 w0 i
∂z
denote the (Reynolds averaged) convergence of subgrid turbulent momentum flux.
All other co-variance terms were found to be negligibly small compared with the
∂
major terms in the equation. ∂x
hu0 u0 i
∂
∂
hu0 v 0 i, ∂y
hv 0 v 0 i
∂x
and
∂
hv 0 u0 i
∂y
are found to be
small and are omitted in Eqn. 3.1 and 3.2 for simplicity. The low-pass fields from
the model are in approximate steady state and we are left with advection, pressure
gradient, Coriolis, and turbulent flux convergence terms. The model response is
very homogeneous in the latitudinal direction because of the assumed structure
of the SST front. It is therefore possible to take an additional average in the
latitudinal direction to increase the statistical stability of the results. Analysis
44
of the remaining terms shows that meridional and vertical advection is small
compared to the other leading terms and we can rewrite the momentum budget
as follows, retaining only the dominant terms.
0 = −u
∂u
1 ∂p
∂
+ fv −
− hu0 w0 i
∂x
ρo ∂x ∂z
(3.3)
0 = −u
1 ∂p
∂
∂v
− fu −
− hv 0 w0 i
∂x
ρo ∂y ∂z
(3.4)
The double overbars for temporal and latitudinal averaging have also been omitted
for simplicity.
In the following illustrations the first three RHS-terms in Eqn. 3.3 and
3.4 are referred to as ‘advection’, ‘coriolis’ and ‘pressure gradient force (PGF)’
terms. The last term is referred to as the turbulent stress convergence term or,
for simplicity, the ‘turbulence’ term. The negative value of advection can also be
interpreted as acceleration and the terminologies will be used interchangeably.
3.2. Results from cold-to-warm SST front experiment (P4)
This section describes the first of two experiments to examine the response
of the atmospheric boundary layer (ABL) to an idealized SST front. The SST
boundary condition for this experiment includes a north-south front in the center
of the inner domain (Fig. 3.1). The background SST from the CAM2 aquaplanet integration is increased by 4 K uniformly along the front to the east
of 60.5◦ longitude. The momentum budgets of the equilibrium and the quasiequilibrium region have similarities and will be discussed together in Section 3.2.1.
The transition region will be discussed in Section 3.2.2.
45
3.2.1. Equilibrium and quasi-equilibrium region (P4)
The equilibrium region is located on the upstream side of the front from the
western boundary of the domain to about 60.5◦ longitude. The quasi-equilibrium
region is located on the downstream side of the SST front from about 61.6◦ to the
eastern end of the domain. The non-equilibrium transition region between 60.5◦
and 61.6◦ longitude is discussed in Section 3.2.2. The air that enters the equilibrium region is in approximate Ekman balance and the advection term is small.
The remaining terms of the momentum budget are approximately constant in the
zonal direction as the air approaches the front near 60.5◦ . The momentum boundary layer top is nearly constant at about 450 m (Fig. 3.2). In the quasi-equilibrium
region located downstream of the SST front, advection again becomes negligible
and the flow has returns to approximate Ekman balance. The momentum balance
in the upstream equilibrium region is consistent with the ABL model of Stevens
et al. (2002). The ABL in the downstream quasi-equilibrium region is slowly adjusting to the new equilibrium condition after passing over the SST front. The
changes agree with the lagged pressure adjustment mechanism found by Small
et al. (2005). Air temperature (Fig. 3.3) changes at a much slower rate than
the SST. Hence, the heat flux associated with the air-sea temperature difference
remains large in the quasi-equilibrium region and therefore the largest pressure
gradients are found there. The slow adjustment process in downstream direction
is shown by the horizontal changes of zonal wind (Fig. 3.7), zonal and meridional
stress convergence (Fig. 3.8) and zonal advection (Fig. 3.10). The zonal and meridional wind velocities are fairly constant at a given height (Fig. 3.7). While the
zonal velocity is larger in the downstream than in the upstream region, the meridional velocity is slightly smaller in the downstream region and the elevated wind
46
maximum is diminished. The increase of zonal velocity is mostly near the surface
which reduces the ABL vertical shear. The convergence of zonal and meridional
turbulent momentum flux (Fig. 3.8) show the adjustment of the ABL towards
Ekman balance for a deepening boundary layer. In the quasi-equilibrium region,
the values of zonal turbulent flux convergence are larger than in the equilibrium
region. In equilibrium conditions, momentum advection is small and vertically
uniform. The vertical structure (see also Fig. 3.30) shows a balance between of
coriolis, advection, turbulence and an increasing PGF term. Therefore, zonal turbulent flux convergence is adjusting to the environment for at least 100 km after
passing the SST front. The meridional turbulent flux convergence term decreases
in the quasi-equilibrium region, but it also shows signs of continued adjustment.
The corresponding turbulence term (Fig. 3.9) shows larger values at a given level
in the downwind region and generally higher values over the warmer SST. As
expected, the meridional stress vanishes near 200 m in the core of the elevated
meridional wind maximum. Meridional stress increases in the downstream region
below the elevated wind maximum. Higher levels show only small changes of
meridional stress.
In the equilibrium region, the ratio of surface stress to boundary layer
height is approximately constant, consistent with the arguments of Samelson et al.
(2006). In the upstream region τs /H = 4.3 × 10−4 s−1 while in the downstream region τs /H = 4.1×10−4 s−1 (Fig. 3.14). This is remarkable, given the 3-dimensional
character of the experiment and the fact that the pressure gradient changes in the
vertical and across the SST-front.
The air temperature increases much slower than the underlying SST. The
maximum air-sea temperature difference and largest sensible heat flux are therefore found downstream of the SST-front. A similar response was observed by
47
Thum et al. (2002) in the equatorial Pacific north of the cold tongue where temperature advection shifts the location of the largest heat flux downwind to the
north of the strongest SST gradients. Since the surface buoyancy and temperature advection are large contributions in the thermodynamic equation, the pressure perturbation is lagged relative to the SST (Fig. 3.11). The resulting zonal
pressure gradient force (Fig. 3.11) is larger in the downstream region and slowly
increases to the eastern edge of the domain. The meridional pressure gradient
force is mostly determined by the large scale pressure distribution that has a
meridional gradient nearly 5 times larger than the zonal component.
Turbulence kinetic energy (TKE) (Fig. 3.12) in the upstream region is
maximum where the shear of zonal and meridional wind is maximum, just below
the elevated meridional wind maximum. Larger values of TKE are found in the
downstream region where two TKE maxima are observed, one near the surface
due to buoyancy production (Fig. 3.13) and one just above the internal boundary
layer. The vertical structure of TKE agrees well with the findings of de Szoeke
and Bretherton (2004) although the pattern of TKE from the WRF simulation is
smoother, possibly because of the long time-averaging period.
Generally, the difference between the upstream and the downstream equilibrium region is most pronounced below 200 m. This observation is another
indication that the vertical resolution near the surface is important.
3.2.2. Non-equilibrium transition region (P4)
The rapidly changing SST in the frontal zone drives significant changes in
the structure of the mean fields and turbulence in the ABL, both horizontally and
vertically. The zonal momentum budget is not in Ekman balance. The part of
48
the boundary layer below 200 m experiences an acceleration over the SST front
(Fig. 3.10). The acceleration is maximum near 61◦ longitude slightly downstream
of the strongest SST gradient. From this point, the near-surface acceleration
gradually decreases, becoming insignificant at about 61.6◦ longitude which marks
the beginning of the quasi-equilibrium region discussed above. The acceleration
is strongest near the surface and confined vertically to below 200m. In an almost
complementary pattern to the acceleration, the zonal turbulent flux convergence
decreases significantly in magnitude below 200m and changes sign near the surface
at around 61◦ longitude (Fig. 3.8). The sign change of the stress convergence is
also found in observational data during EPIC 2001 C130 flights over the equatorial
SST front (Small et al. 2005, their Fig. 13c) and is similar to the LES model results
of de Szoeke and Bretherton (2004). The negative values aloft are slightly shifted
downstream relative to the near-surface positive values.
The values of the pressure and Coriolis terms in this region rapidly approach their values in the downstream equilibrium region, but the changes in
magnitude are much smaller compared to the advection and turbulence terms,
especially near the surface.
3.2.3. ABL average and surface momentum budget (P4)
As a result of the vertical dipole structure of the zonal turbulent stress,
the ABL-averaged momentum budgets (Fig. 3.15) are significantly different than
the near-surface momentum budgets in the transition region. The ABL-averaged
budget shows that the zonal and the meridional flow is close to Ekman balance
with relatively small accelerations in the zonal direction. The zonal and meridional residual from the sum of advection, PGF, coriolis and turbulence terms
49
are denoted as X and Y , respectively. The zonal residual X and the meridional
residual Y can be interpreted as the sum of effects from entrainment, changes in
boundary layer depth and other subgrid-scale mechanisms such as the effects of
shallow convection on the ABL-averaged momentum budget. The residuals become significantly less than zero in the region where the boundary layer deepens,
consistent with the notion of entrainment by changes in the boundary layer depth.
In the simplest form, entrainment may be parameterized following Stevens et al.
2001 who use a bulk entrainment to define the stress at the top of the momentum
boundary layer
τ = we ∆U = we (UT − U)
(3.5)
where we is the entrainment velocity and UT is the vector wind above the ABL.
Note that the residuals X and Y and entrainment have opposite signs due to
the definition of X and Y as a residual. Changes in pressure gradient force are
compensated by changes in the Coriolis term.
In contrast, the surface momentum budget (Fig. 3.16) is highly
ageostrophic in the transition region. The dominant balance of the zonal budget
is between the advection and stress convergence terms. This zonal and meridional
force balance is most easily observed in vector form (Fig. 3.17). Near the surface,
the vector of turbulent stress convergence rotates clockwise where zonal advection
becomes important. For a brief period, the zonal component of the turbulent
stress vector is pointing in the direction of the zonal wind. In the transition region, turbulent stress convergence decreases, and in parts of the transition region,
momentum stress convergence is accelerating the flow. The changes in the PGF
and Coriolis terms are relatively small. The meridional surface budget terms are
50
larger in magnitude but the transition zone changes are smaller. The vertically
integrated ABL seems to adjust more geostrophically than the near-surface layers.
3.3. Results from warm-to-cold SST front experiment (M4)
The M4 experimental setup is similar to the P4 simulation; only the SST
boundary condition changes. Instead of a positive 4K change, the sea surface
temperature is decreased by 4 K in the eastern part of the domain to examine the
adjustment of the ABL to decreasing SST. The prevailing wind direction is now
towards decreasing SST. As in P4, short periods with easterly wind directions
are removed from the average, although no qualitative changes were found when
retaining the entire record. The transition zone is now characterized by the sudden
collapse of the momentum boundary layer to below 200 m with a subsequent slow
deepening to around 230 m in the downstream quasi-equilibrium region. We again
divide the domain into upstream equilibrium, transition, and downstream quasiequilibrium regions based on the importance of the advection term (Fig. 3.18).
The locations of the boundaries of the transition region are similar in the two
experiments. In the M4 experiment, the downstream boundary of the transition
region has moved slightly upstream, making the transition region slightly smaller
in the M4 experiment.
3.3.1. Equilibrium and quasi-equilibrium region (M4)
The location of the equilibrium region boundary in M4 coincides with that
of the P4 run. The quasi-equilibrium region, however, begins about 0.4◦ of longitude closer to the SST-front near 61.2◦ longitude. The decrease of ABL height
between the upstream equilibrium and the downstream quasi-equilibrium region
51
is more than 200 m, essentially a collapse of the initial convectively driven momentum boundary layer. Higher zonal velocities are observed in the equilibrium
region compared to the quasi-equilibrium region (Fig. 3.19). The differences between equilibrium and quasi-equilibrium region are most pronounced near the
surface where the zonal wind decreases by 3 ms−1 . In the horizontal, there is
very little zonal wind speed change within each region.The meridional velocity
shows an increase in the elevated wind maximum in the quasi-equilibrium region.
The near-surface northerly meridional wind becomes stronger by about 1 ms−1 .
The vertical shear for both meridional and zonal wind increases in the quasiequilibrium region. The vertical convergence of zonal turbulent stress (Fig. 3.20)
is almost identical in the equilibrium and quasi-equilibrium regions. However,
meridional stress convergence is significantly larger in the quasi-equilibrium region. The decrease of zonal turbulent stress (Fig. 3.21) in the quasi-equilibrium
region reflects the deceleration of zonal wind compared to the equilibrium region.
The zonal advection (Fig. 3.22) is small above the boundary layer in the quasiequilibrium region. However, in the layer that was detrained from the boundary
layer there is negative advection which is consistent with the notion that the detrained air is able to accelerate freely until a new momentum balance is established
(de Szoeke and Bretherton 2004). The meridional advection term is larger in the
quasi equilibrium region, which is related to the strengthening of the elevated
meridional wind maximum. The cooling of the air over the cooler SST leads to
hydrostatic increases in pressure in the quasi-equilibrium region and the zonal
pressure gradient (Fig. 3.23) changes accordingly with most significant changes
in the boundary layer. The meridional pressure gradient is mostly controlled by
the large-scale forcing. Turbulence kinetic energy has a similar structure in the
upstream and downstream equilibrium regions. The maximum in TKE is found in
52
the lower third of the boundary layer. The ratio of surface stress to boundary layer
height is not equal in the equilibrium and quasi-equilibrium region (Fig. 3.25). In
the upstream region, we find τs /H to be 4.5 × 10−4 s−1 , while in the downstream
region this ratio is closer to τs /H = 5.6 × 10−4 s−1 . The increase in τs /H between
the equilibrium and quasi-equilibrium region can be explained by the increase in
pressure gradient which should translate into higher surface winds. This indicates
that changes in surface pressure across the SST front are important.
3.3.2. Non-equilibrium advection region (M4)
The transition region is characterized by the sudden collapse of the boundary layer as the SST begins to decrease. The most significant changes in zonal
surface wind (Fig. 3.19) are observed in this region. The changes at a constant
elevation within the upper part of boundary layer are much weaker. The meridional wind remains virtually constant horizontally and vertically. The momentum
boundary layer collapse is most easily observed in the longitude-height crosssections of zonal turbulent flux convergence (Fig. 3.20), which is confined to a
very shallow layer. The meridional flux convergence increases toward the surface
without reaching an extremum. The zonal stress (Fig. 3.21) decreases rapidly
while meridional stress only decreases slightly. The decrease of zonal momentum
(Fig. 3.22) is observed only below 100 m. The meridional advection in this region
shows only small differences from the upstream and downstream regions but is
decelerating in the transition zone. No sudden changes in the pressure gradient
response can be observed (Fig. 3.23). However, the zonal pressure gradient decreases faster in the boundary layer than above. At a given height within the
boundary layer, turbulence kinetic energy decreases and is fairly homogeneous
53
almost throughout the entire boundary layer. The fact that the ratio of surface
stress to boundary layer height is not constant (Fig. 3.25) is an indication that
the boundary layer height in the transition region is either overestimated or that
other processes in the transition zone are of equal importance.
3.3.3. ABL average and surface momentum budget (M4)
The boundary layer averaged momentum budget (Fig. 3.26) shows that
the integrated boundary layer flow is approximately in Ekman balance in the
transition zone. The change of the zonal pressure gradient is mostly balanced by
a change in Coriolis force and a small reduction of turbulent stress. The residual
of Coriolis, advection, turbulent stress convergence and pressure gradient force
is interpreted as a detrainment process and is positive in accordance with the
decrease in boundary layer height. The change of the Coriolis force across the
SST-front is similar in the surface (Fig. 3.27) and the ABL-average (Fig. 3.26)
momentum budgets. The similarity between surface and ABL-average is also
found in the PGF term. However, zonal momentum stress convergence and zonal
momentum advection show substantial differences between near-surface and the
ABL-average response. The momentum budget terms are shown in vector form
to illustrate the force balance (Fig. 3.28). In the transition zone the near-surface
acceleration becomes important and the stress convergence vector rotates counterclockwise before adjusting to the new Ekman balance in the quasi-equilibrium
region.
54
3.4. Comparison of the responses between P4 and M4
The qualitative and quantitative differences in the adjustment mechanism
of P4 and M4 are highlighted by adding the two simulations and removing the
zonal average from the sum. If the responses are completely symmetric, the composites will be zero. Any non-symmetric responses will deviate from zero. The
results show that asymmetries are found mostly in the transition zone (Fig. 3.29).
The largest differences occur near the upstream boundary of the transition region while smaller differences are found closer to the downstream boundary of
the transition region. The large asymmetries near 60.9◦ longitude reflect the fact
that the deceleration and flux convergence in the M4 transition zone are confined
below 100 m while the acceleration and flux divergence in the P4 transition region
extend to around 200 m. The small asymmetry near the downstream boundary of
the transition zone around 60.7◦ longitude is the result of the slight upwind shift
of the advection and flux convergence terms in M4 relative to the acceleration and
flux convergence terms in P4. The momentum flux composite and the advection
composite are almost complementing one another except for some stress convergence near 61.2◦ longitude between 300 m and 400 m. The differences between the
P4 and M4 transition zones are a consequence of the different adjustment mechanism between the warm-to-cold and cold-to-warm flow. In the warm-to-cold (M4)
case, the boundary layer collapses suddenly and the deceleration takes place over
shorter spatial scales than the acceleration associated with the convective mixing
in the internal boundary layer in the cold-to-warm (P4) case.
55
3.5. Discussion
Several mechanisms have been proposed to explain the strength and the
location of the acceleration of near-surface winds in the vicinity of oceanic SST
fronts. We presented simulations of the WRF regional mesoscale model with high
vertical resolution in the boundary layer to analyze the importance of the various
possible mechanisms for high correlation between mesoscale anomalies of surface
wind stress and SST in the vicinity of large SST gradients. The importance of
stability-induced changes in turbulent momentum transport has been emphasized
by a number of studies (Sweet et al. 1981, Wallace et al. 1989, Park et al. 2002,
and de Szoeke and Bretherton 2004). Mahrt (1972) and Tomas et al. (1999) have
noted that momentum advection in flows in low latitudes is important. Other
studies (Mahrt et al. 2004, Cronin et al. 2003, Small et al., 2005) have suggested
that a (lagged) pressure adjustments is responsible for the SST-wind correlation.
Samelson et al. (2006) argue that surface stress depends on the boundary layer
depth. This numerical study includes pressure as a state variable and resolves
the boundary layer in adequate detail to discuss the importance of pressure adjustment, momentum advection, turbulent stress convergence and the changes in
boundary layer height.
The near-surface wind acceleration and deceleration is found in a narrow
transition zone of about 100 km co-located with the SST-front. In this zone, we
find that the change of pressure gradient is small. Therefore, the lagged pressure
adjustment mechanism by Cronin et al. (2003) and Small et al. (2005) cannot
explain the acceleration and deceleration in the vicinity of the SST-front. In
the quasi-equilibrium region downstream from the transition region, an Ekman
56
balance is found between pressure gradient force, turbulent stress convergence and
Coriolis force.
The balance between the pressure gradient and frictional forces alone cannot explain the vertical differential advection. While the pressure gradient force
(Fig. 3.30, Fig. 3.31) is very similar below and above 200 m, the advection and
stress convergence change dramatically in the lower 200 m. The large vertical
changes and the shallowness of the internal boundary layer might also explain
why other coarser vertical resolution models have had difficulties in simulating the
observed stress convergence in the transition zone, keeping in mind that turbulent
stress is modeled as a derivative quantity and hence turbulent stress convergence
as a second derivative. Our findings, which are consistent with satellite observations (Chapter 1) and with observations of the marine boundary layer (Friehe
et al. 1991), demonstrate the importance of adequate vertical resolution of the
boundary layer. With coarser vertical resolution in the boundary layer used by
Small et al. (2005) and by global circulation models it is not possible to simulate
important stress convergence features near the surface.
We have shown that the quantity τs /H is not constant across step changes
of SST. In the far fields where advection becomes small and τs /H can be obtained
by integration, we have shown that pressure anomalies are as important as changes
in boundary layer depth. In the transition zone where pressure gradient force
might be assumed constant with height, we have found that advection is important
and cannot be ignored. Therefore, the assumptions leading to τs /H = constant are
not applicable to at least part of the transition region. Indeed, we have presented
a simulation where turbulent stress convergence is acting to accelerate the surface
winds in the downstream direction.
57
Our results suggest that entrainment of momentum by a convectively
driven thermal internal boundary layer (TIBL) is the mechanism controlling the
acceleration and the strength of the turbulent flux convergence in the transition
zone of the cold-to-warm experiment. A TIBL has been found by Friehe et al.
(1991) in a model simulation of the Frontal Air-Sea Interaction Experiment. The
TIBL also determines the strength of the pressure perturbations since initially the
air only moistens and warms within the TIBL. From the sequence of profiles in
the transition region (not shown), it becomes apparent that the response of turbulence and advection somewhat leads the response of the pressure gradient. This
suggests that the pressure gradient is not involved in determining the location of
the acceleration in the transition zone.
Turbulent flux convergence can act to accelerate the flow in two ways, 1) by
simply reducing the vertical convergence of the stress that usually acts to decelerate the flow, or 2) by actively accelerating the flow in the case when momentum is
redistributed in the boundary layer. The TIBL is important in both cases. Once
the TIBL forms at the SST-front, an intricate new balance between turbulence,
pressure gradient and advection develops. The large sea-minus-air potential temperature difference triggers the formation of a convectively driven internal boundary layer which then grows by entrainment. Entrainment incorporates higher
velocity air into the TIBL, while at the same time enhanced mixing reduces the
vertical shear, thus increasing stress while decreasing stress convergence. The reduced stress convergence is balanced by an acceleration of the near-surface wind.
Advection increases vertical shear while turbulent mixing reduces it. Advection
and turbulent flux convergence may strengthen simultaneously until convectively
driven entrainment and mixing can no longer be maintained or becomes too weak.
The wind shear is large in the entrainment zone of the internal boundary layer.
58
Therefore, and in combination with large mixing coefficients, turbulent fluxes increase aloft and become larger than the surface flux. The near-surface turbulent
stress convergence becomes positive and turbulent flux actively accelerates the
flow.
The stabilizing effects of decreasing SST in the downwind direction in the
M4 simulation efficiently cuts off the buoyancy generation of TKE that supports
the boundary layer. As a result, the momentum boundary layer collapses and
little mixing remains above the new boundary layer top at about 200 m. The
air aloft is free to accelerate which effectively separates faster moving air from
the slower moving air within the boundary layer. The pressure gradient in the
new shallow boundary layer cannot support the upstream wind speed and the
reduced mixing acts to decelerate the surface winds. The resulting deceleration
leads to strong advection near the surface and reduces the vertical shear. As in the
P4 simulation, advection and turbulent stress convergence increase in magnitude
simultaneously to efficiently reduce surface wind speed and adjust the flow towards
the new balanced state.
59
FIGURE 3.1. Maps of background SST in the region of the Agulhas return current. Shaded SST is shown in 2◦ C contour intervals. Gray rectangles denote the
boundaries of the 3 nested WRF domains. Inside the WRF domains the SST is
increased by 4 K within a narrow north-south frontal region.
60
FIGURE 3.2. Bin-averaged distribution of normalized momentum boundary layer
depth as function of longitude in the upstream region. Each vertical profile is
a distribution at a given longitude. A thick line denotes the ABL depth that
occurred most often. Top panel is derived from the P4 experiment and the bottom
panel from the M4 experiment. The vertical bin size for each distribution is 50m.
Shown is the region upstream of the SST front where the momentum boundary
layer depth is not influenced by the SST front.
61
FIGURE 3.3. Longitude-height cross-section of the potential temperature θ. Top
panel shows results from P4 and bottom panel results from M4. The solid line in
each panel denotes the momentum boundary layer depth; the thick dashed line in
the upper panel is a measure of the internal boundary layer depth. See text for
the definitions of the boundary layer depths.
62
FIGURE 3.4. Sea surface temperature at 43◦ S as function of longitude. Dots
represents model gridpoint locations. Latitudinal dependence of SST is approximately a cosine function of the latitude.
63
FIGURE 3.5. Top: Near-surface advection as a function of longitude. Bottom:
Co-located sea surface temperature distribution as function of longitude.
64
Thermodynamic boundary layer depth
Momentum boundary layer depth
less stable
weakly stable
Entrainment Zone
unstable
Equilibrium Region
Transition Region
Quasi−Equilibrium Region
FIGURE 3.6. Schematic description of the cold-to-warm experiment.
65
FIGURE 3.7. As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
velocity of the P4 experiment.
66
FIGURE 3.8. As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
turbulent stress convergence of the P4 experiment.
67
FIGURE 3.9. As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
turbulent stress of the P4 experiment.
68
FIGURE 3.10. As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
advection of the P4 experiment.
69
FIGURE 3.11. As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
pressure gradient force of the P4 experiment.
70
FIGURE 3.12. As in Fig. 3.3, except for the turbulent kinetic energy of the P4
experiment.
71
FIGURE 3.13. As in Fig. 3.3, except for the buoyancy production of turbulent
kinetic energy of the P4 experiment.
72
FIGURE 3.14. Ratio of surface stress to momentum boundary layer depth as
function of longitude. Solid line denotes the magnitude τs /H, dashed the zonal
component τx /H and dotted the meridional component τy /H.
73
FIGURE 3.15. Momentum boundary layer average zonal (top) and meridional
(bottom) momentum budget as function of longitude. The zonal and the meridional residual is denoted by X, Y, respectively.
74
FIGURE 3.16. Zonal (top) and meridional (bottom) momentum balance at near
surface model level (12m) as function of longitude. Shown are advection (solid),
coriolis (thick dash-dotted), pressure gradient (dashed) and turbulent stress convergence (thick dotted).
75
FIGURE 3.17. Vertically staggered vector representation of the momentum budget from P4. Northward vectors are pointing up and eastward vectors point to
the right. Top panel vectors represent Coriolis (orange), advection (blue), turbulent stress convergence (“τ ”, black), and pressure gradient force (“PGF”, green).
Additionally wind direction (“Wind”) is overlaid in gray. Bottom panel shows
selection of top panel magnified with colored vectors for advection (blue) and
turbulent stress convergence (black), other vectors are repeated for completeness,
but grayed-out for clarity. Note that not all grid locations are showed.
76
FIGURE 3.18. Top: Near-surface advection as a function of longitude. Bottom:
Co-located sea surface temperature distribution as function of longitude.
77
FIGURE 3.19. As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
velocity for the M4 experiment.
78
FIGURE 3.20. As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
turbulent stress convergence of the M4 experiment.
79
FIGURE 3.21. As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
turbulent stress of the M4 experiment.
80
FIGURE 3.22. As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
advection of the M4 experiment.
81
FIGURE 3.23. As in Fig. 3.3, except for the zonal (top) and meridional (bottom)
pressure gradient force of the M4 experiment.
82
FIGURE 3.24. As in Fig. 3.3, except for the turbulent kinetic energy of the M4
experiment.
83
FIGURE 3.25. Ratio of surface stress to ABL depth as function of longitude.
The solid line denotes the magnitude τs /H, dashed the zonal component τx /H
and dotted the meridional component τy /H.
84
FIGURE 3.26. Momentum boundary layer average zonal (top) and meridional
(bottom) momentum budget as function of longitude. The zonal and the meridional residual is denoted by X, Y, respectively.
85
FIGURE 3.27. Zonal (top) and meridional (bottom) Momentum Balance at near
surface model level (12m) as function of Longitude. Shown are Advection (solid),
Coriolis (thick dash-dotted), Pressure gradient (dashed) and turbulent stress convergence (thick dotted).
86
FIGURE 3.28. Vertically staggered vector representation of the momentum budget from M4. Northward vectors are pointing up and eastward vectors point to
the right.Top panel vectors represent Coriolis (orange), advection (blue), turbulent stress convergence (“τ ”, black), and pressure gradient force (“PGF”, green).
Additionally wind direction (“Wind”) is overlaid in gray. Bottom panel shows
selection of top panel magnified with colored vectors for advection (blue) and
turbulent stress convergence (black), other vectors are repeated for completeness,
but grayed-out for clarity. Note that not all grid locations are shown.
87
FIGURE 3.29. Longitude-height cross section of 1) zonal turbulent flux convergence from P4 and M4 added and zonal mean removed (top) and 2) zonal
advection from P4 and M4 added and zonal mean removed (bottom).
88
FIGURE 3.30. Vertical profile of the zonal (top) and meridional (bottom) momentum budget from P4 at three locations. Shown from left to right are the
upstream, transition and downstream region relative to the SST front. Note the
order-1 difference of scales of zonal and meridional budgets.
89
FIGURE 3.31. Vertical profile of the zonal (top) and meridional (bottom) momentum budget from M4 at three locations. Shown from left to right are the
upstream, transition and downstream region relative to the SST front. Note the
order-1 difference of scales of zonal and meridional budgets.
90
4. ABL RESPONSE TO IDEALIZED MESOSCALE
MIDLATITUDE SST ANOMALIES
In the Agulhas region, the monthly averaged SST pattern appears to be
a quasi-periodic series of warm and cold anomalies with spatial scales on the the
order of 200-500 km (Fig. 2.1), giving an ABL horizontal advective timescale of
∼ 6 hours. These anomalies are found over the world ocean and suggests that, in
many regions with large SST variations, the ABL must be continually adjusting
to changing SST. Before the downwind quasi-equilibrium conditions simulated in
Chapter 3 can occur, another adjustment process has begun for the next SST
anomaly. In this chapter we therefore extend our study of ABL adjustment to
more realistic SST perturbations that are similar to the SST perturbations of the
Agulhas region (Fig. 4.1). To remove ambiguity in the definition of the spatial
scale, we examine the ABL response to sinusoidal SST anomalies. The budgets of
heat and momentum then can be related directly to the SST perturbations and
the phase relationships among the dependent variables can be studied. The periodic nature of the assumed SST pattern also facilitates a more robust statistical
analysis.
Although vertical redistribution of momentum is important for the wind
accelerations in the lower ABL, we will see that vertical momentum exchange is
not the driving mechanism for the wind stress response. We show how the heating
pattern creates pressure perturbations that can drive the acceleration of surface
winds and we speculate how this adjustment changes with changing scales of the
SST perturbations. We use the ratio τs /H of surface stress and boundary layer
depth as an indicator to determine if the boundary layer is in a simple Ekman
balance or if additional terms are necessary to describe the adjustments.
91
This chapter is organized in the following way: In the first section, the
model configuration, the initial model fields and boundary conditions, including
SST forcing, are briefly described. Section 4.1 also describes the methodology of
the analysis and discusses the major force balances of the surface and the ABLaveraged momentum budget. Section 4.2 shows the importance of the pressure
gradient force as the driving mechanism of the observed near-surface wind adjustments and demonstrates the good agreement between simulated surface wind
divergence term and the horizontal Laplacian of the pressure term in the divergence equation.
4.1. Model description and initial fields
We analyze the ABL response to a field of mesoscale SST anomalies arranged in a checkerboard pattern with scales similar to those observed in the
Agulhas region. The basic model setup is identical to Chapter 3 except for the
underlying SST and the number of nested domains. The initial atmospheric fields
and boundary conditions were obtained from a symmetric aqua planet CAM2
simulation performed by Eric Maloney. The CAM2 simulation was spun up for
3 years. The 30 consecutive days after spin up were used to initialize the WRFmodel and to obtain the lateral boundary conditions. For practical reasons, the
intermediate nest was dropped in favor of a larger inner domain with 20◦ ×8◦ grid
size. The grid spacing of the inner domain is 15 km, which is roughly two times
larger than in the SST-front experiments considered in Chapter 3, but is adequate to resolve the mesoscale SST anomaly pattern. The high vertical resolution
of the model experiment in Chapter 3 was maintained because of its importance
for resolving the large vertical differential advection of horizontal momentum and
92
stress convergence (compare Fig. 3.30, Fig. 3.31, for example). The outer to inner
grid ratio of 5:1 does not noticeably change the behavior of the model near the
lateral boundaries compared to the 3:1 ratio in Chapter 3. However, some noise is
apparent near the boundaries of the inner nest (Fig. 4.2) which cannot be removed
by the 2-D loess filtering used here. Irrespective of the origin of the noise, no data
within 1/2◦ of the inner nest boundary was used in the analysis. The first 2 days
of simulation are discarded to allow the model some adjustment time. The mean
wind direction over the 28 days is west-north-west, close to the zonally symmetric
flow from the CAM2 aqua-planet simulation. The 1-month averaging period is not
sufficiently long to remove all synoptic variability that causes departures from the
zonal flow. The integration period includes short intervals with prevailing easterly
surface wind. As in Chapter 3, these easterly wind periods were excluded from
the analysis. Note that the results remain qualitatively the same when including
the periods of easterly wind. However the magnitude of the averaged response is
reduced. Model output is composited relative to the SST anomalies so that we
may interpret the ABL response as the result of prevailing westerly flow over a
periodic series of warm and cold SST anomalies.
4.1.1. SST Forcing
The SST pattern is derived by estimating the spatial scales and magnitude
of the SST variations in the Agulhas region (Fig. 4.2). The dominant scales are
approximated as 4◦ longitude and 5◦ latitude with amplitudes of 2.5 K. These
values were used to generate SST perturbations that vary in meridional and zonal
direction as cosine functions of the latitude and longitude, respectively. The
resulting SST field used to perturb the aqua planet background SST is shown in
93
(Fig. 4.1). Near the strongest SST gradients, the SST changes by about 4 K over
a distance of about 1◦ longitude. The maximum SST gradient is thus about 1/2
of the values used in the SST front experiment in Chapter 3.
4.1.2. Results
We calculate perturbations of the flow by applying a spatial high-pass filter to all meteorological variables. We use the same 30◦ longitude by 10◦ latitude
loess-filter that was discussed in Chapter 2. The similarity of the response of
the zonal and meridional wind perturbations over the entire inner domain for
all positive anomalies, and for all negative anomalies (Fig. 4.2), motivates compositing the ABL response relative to the SST perturbations. The homogeneity
of the response also allows us to examine the variability in the zonal direction
by averaging the meteorological fields in meridional direction only. The resulting
height-longitude and surface composite figures cover a spatial area that extends
2.5◦ or 1/2 wavelength of the SST perturbation in meridional direction and the
entire domain in zonal direction.
For consistency and to simplify comparison between this and the previous
results in Chapter 3, the methodology from Chapter 3 is used to determine the
average momentum boundary layer depth. Since the previous simulations and this
simulation are based on the same aqua planet boundary conditions, it is reasonable
to assume the same momentum boundary layer depth of 450 m over unperturbed
SST. The value of turbulent flux convergence that represents the boundary layer
depth (S450 ) is assumed to be 2 × 10−4 ms−2 . The exact value that is chosen
to represent the momentum boundary layer depth is not crucial for the analysis
94
in this chapter. The conclusions of this chapter are the same for other values or
methods (not shown) to determine the momentum boundary layer depth.
Although the analysis will focus mainly on the composited fields, the spatial relation between the SST perturbation and the ABL response can more easily
be perceived initially by eye by visualizing the response over the entire domain.
The maps of near-surface zonal and meridional velocity (Fig. 4.2) reveal the consistency of the ABL response to the SST perturbation. The wavelength of the SST
perturbation is used as the unit length of phase (= 360◦ ) to quantify the phase relation between wind and SST perturbations. It is apparent that the zonal velocity
is phase locked to the SST perturbations with a shift of about 30◦ longitudinally
and 7◦ latitudinally to the south-east relative to the SST perturbation. The amplitude of the zonal velocity perturbation is about 2 ms−1 . The meridional velocity
anomaly is phase shifted approximately 90◦ to the east and is nearly in phase in
the south-north direction relative to the SST anomaly. Therefore, meridional velocity is in phase with the SST gradient. The meridional velocity anomalies are on
the order of 0.6 ms−1 , about one-third weaker than the zonal velocity anomalies.
The spatial scale of the imposed SST perturbations is different in the meridional
and zonal direction, thus creating smaller SST gradients in the meridional direction than in the zonal direction. The reasons for the phase relationships will be
addressed at a later point but the horizontal scale difference explains the differences in the zonal and meridional velocities. The slight off-meridian location of
the zonal velocity pattern towards the south-east relative to the SST perturbation
agrees very well with the direction of the prevailing wind.
The longitude-height cross sections in Figs. 4.3 – 4.6 are used to show
the vertical structure of quantities that are important for the boundary layer
momentum budgets. Each cross section shown is chosen along the latitude of
95
maximum ABL response to the SST anomalies for a given variable. The lower
panel of Fig. 4.3 shows the vertical structure of zonal wind and turbulent stress
convergence in relation to the boundary layer depth. The top panel in Fig. 4.3
shows SST, boundary layer depth, and perturbations of near-surface advection,
turbulent stress convergence, coriolis force, and pressure gradient force at the same
latitude of maximum ABL response.
The response of zonal wind is strongest near the surface and becomes small
above 120 m Fig. 4.3. The zonal turbulent stress convergence term has a dipole
pattern with largest values near the surface and a reversal in sign at around 120 m.
A similar pattern was found in the transition zone over the SST fronts Chapter
3. While the stress divergence maximum generally is observed over negative SST
gradients, the precise location of the maximum turbulent stress divergence is exactly where the momentum boundary layer depth minimum occurs. Maximum
turbulent stress convergence, on the other hand, occurs over the maximum positive SST gradients. The amplitude of the surface zonal stress convergence is about
2. × 10−4 ms−2 , which is comparable to the amplitude found in the P4 and M4
experiment.
The growth and collapse of the momentum boundary layer as air flows
over the SST anomalies is apparent in the middle and upper panels of Fig. 4.3.
The change of H is not symmetric relative to the SST. The growth phase from
the minimum to maximum momentum boundary layer depth is almost 3 times
as long as the short collapsing phase. The maximum H is shifted about 25◦ in
wavelength downstream in terms of SST phase while the minimum H is observed
35◦ in wavelength upstream of the SST minima. In regions of surface cooling,
the momentum boundary layer collapses more rapidly than it increases over regions of surface warming. The momentum boundary layer begins to deepen in
96
a region where downwind SST is still decreasing (e.g. near 67◦ longitude) and
it continues to grow in regions where the downwind SST is already decreasing
(e.g. 70.1◦ longitude). The buoyancy flux, generated by the air-sea temperature
difference, heats the boundary layer from below and generates turbulence. Once
the heating is cut off, the turbulence decays rapidly in the layer above. On the
other hand, it takes time to deepen the boundary layer by buoyancy because turbulent elements have to overcome the slightly stable environment. Over time, the
surface heating therefore gets distributed over a deeper layer.
Samelson et al. (2006) note that the ratio τs /H of surface stress and boundary layer depth should be approximately constant under the assumption of constant pressure gradients and equilibrium conditions. Greater surface wind stress
should therefore be associated with a deeper momentum boundary layer. If τs /H
is not constant, the ABL may not be in equilibrium or other effects that were neglected may be important. The ratio τs /H therefore may be used as an indicator
of where a simple steady Ekman balance does or does not hold. The ratio may
be used to divide the third panel into 3 distinct sections: (1) a section of fairly
constant ratio of around 3 × 10−4 s−1 , (2) a section of rapid increase in τs /H to as
much as 7 × 10−4 s−1 , and (3) a subsequent section of slow decrease back to the
constant value (Fig. 4.3, third panel).
The two dominant terms of the surface zonal momentum budget – zonal
momentum advection and stress convergence – are shown in second panel of
Fig. 4.3. Similar to the SST front experiment in Chapter 3, the two terms have
the same phase relationship to the SST and are nearly a mirror-image of the other:
positive advection and turbulent flux divergence over negative SST gradients and
vice versa.
97
Similarly Fig. 4.4 shows a longitude-height cross section, but for meridional
velocity and meridional turbulent convergence. The pattern of turbulent stress
convergence tilts eastward while the contours of meridional velocity are tilted
westward with height. The maximum meridional wind is not found near the
surface, but aloft at around 120 m.
The composites of the zonal and meridional surface pressure gradient perturbations show that the zonal surface pressure gradient perturbation nearly in
phase with the SST perturbation (Fig. 4.5). Assuming hydrostatic balance, it
might be expected that pressure anomalies are in phase with SST anomalies and
the pressure gradient force therefore should be in phase with the SST gradient.
However, temperature advection (Thum et al. 2002; Small et al. 2004) and the
delay in upper-layer heating shifts the pressure gradient downwind. In a vertically
integrated sense, the surface pressure gradient force can be approximated following
Mahrt et al. (1982) and assuming no contribution from the free atmosphere:
∂H
∂θ
1 ∂P
= −g∆θ
+ ΓH ,
ρo ∂x
∂x
∂x
g
where Γ = .
θo
−
(4.1)
(4.2)
The first term on the RHS corresponds to the pressure gradient force in the
shallow water equations accounting for a change in the ABL depth H and the
second term accounts for the change in perturbation pressure gradient force due
to horizontal variation in ABL density. The two terms usually have opposing
effects: As the boundary layer deepens, less dense air above the boundary layer is
replaced by denser boundary layer air, thereby creating a pressure gradient force
towards the region with lower boundary layer depth in analogy to the shallow
water equation. The second term, however, creates a pressure gradient force in the
direction of increasing boundary layer depth because the heat flux decreases the
98
average density of the boundary layer. Cronin (2003) suggested that both terms
balance in the region where the boundary layer deepens rapidly, thereby shifting
the response of the pressure gradient. In this experiment, however, the weak
stratification near the boundary layer top creates only small density differences
and the effect of the shallow water term is negligible. The pressure gradient force
can be estimated by the change in potential temperature alone and Eqn. 4.1 can
be written as:
−
1 ∂P
∂θ
≈ ΓH ,
ρo ∂x
∂x
(4.3)
The model pressure gradient force and the estimated pressure gradient force based
on Eqn. 4.3 show a good agreement in terms of magnitude and location relative
to the pattern of SST (Fig. 4.6).
The surface momentum budget is not in Ekman balance (Fig. 4.4). Zonal
advection is balanced by stress convergence and a significant ageostrophic component is observed because the zonal Coriolis force and the pressure gradient force
show a similar rather than opposing surface pattern. The resulting ageostrophic
wind is important for maintaining the balance of the surface zonal momentum
budget.
The largest term of the meridional momentum budget over maximum SST
anomalies is the meridional Coriolis force (Fig. 4.4). It originates from the large
changes of the zonal velocity. The meridional pressure gradient is negligible at the
latitude of strongest SST anomalies (Fig. 4.5), leaving the coriolis term unbalanced
by the pressure gradient force. The balance is maintained by the meridional
component of the stress convergence and meridional advection of momentum.
Although the surface pressure gradient is in phase with the SST anomalies in the
meridional direction, the meridional surface stress is not in phase with the SST
99
pattern but is shifted about 30◦ in wavelength to the south relative to the SST
anomalies. This phase shift might be explained by the large Coriolis term right
over the maximum SST anomalies.
Assuming zero stress at the top of the momentum boundary layer, the
monthly layer-averaged stress convergence is expected to equal the negative of
the surface stress written in a bulk formulation τ (H) − τ (0) = H ∗ Cd ∗ v ∗ |V | .
This is true for the zonal component of stress and surface wind. However, we find a
discrepancy between the monthly layer-averaged meridional stress and the meridional surface stress from the bulk formula. This indicates that the nonlinearities
in the bulk formulation are important.
4.2. Discussion
The findings of this chapter differ in several aspects from the expectations derived from simple quasi 2-dimensional flow across an SST front considered
in Chapter 3. The growth and decay of the boundary layer and the significant
changes in τs /H clearly indicate that a simple balance between stress convergence
and a change in boundary layer depth as suggested by Samelson et al. (2006) is
not adequate on this scale. The pressure is not constant across the SST anomalies. The scale of the SST pattern is such that the surface heating persists long
enough so that it can heat a deep enough layer to change the pressure significantly. Temperature advection creates a shift in the pressure response. The scale
of the response of the surface stress is similar to the scale of the pressure response. Changes in pressure, friction and momentum are important components
of the momentum budget.
100
An important conclusion of Samelson et al. (2006) is that in the vertically
averaged momentum boundary layer budget the stress convergence becomes τs /H
and cannot drive the near-surface acceleration. For the zonal layer-averaged momentum budget, we find a balance between the stress convergence, the pressure
gradient force, zonal advection and coriolis force (Figs. 4.7 – 4.10). The pattern of
layer-averaged stress convergence and pressure gradient force are similar (Fig. 4.7
and Fig. 4.8) which indicates that the layer-averaged pressure is the driving force
on this scale. We have also shown that the pressure gradient force can be estimated by the gradient of the perturbation potential temperature (Eqn. 4.3), which
in turn can be related, as we shall see in Chapter 5, to the SST perturbation by a
simplified thermodynamic energy equation. The layer-averaged equations are approximately linear and may be relatively easy to solve and interpret. We present
such an analysis in Chapter 5.
101
FIGURE 4.1. Maps of background SST in the region of the Agulhas return current. Shaded SST is shown in 2◦ C contour intervals. Gray rectangles denote the
boundaries of the 2 nested WRF domains. Contour interval for the WRF domains
is 1◦ C, outside contour interval 2◦ C
102
FIGURE 4.2. Map of near-surface zonal (top) and meridional (bottom) wind
speed perturbations with SST contour and mean wind speed vector overlay.
103
FIGURE 4.3. Composites of boundary layer variables as function of longitude:
SST perturbation and boundary layer depth (top panel) near the location of
maximum boundary layer depth variation. Perturbation of advection and turbulent stress convergence (second panel). Ratio of perturbation surface stress and
boundary layer depth (“τ /H”), Coriolis force (“Coriolis”) and pressure gradient
force (“PGF”) (third panel). Longitude-height cross section (bottom panel) of
meridional turbulent stress convergence (shaded) with zonal velocity (black) and
boundary layer depth (gray) contours overlaid.
104
FIGURE 4.4. Composites of boundary layer variables as function of longitude:
SST perturbation and boundary layer depth (top panel) near the location of maximum boundary layer depth variation. Perturbation of advection and turbulent
stress convergence (second panel). Coriolis force (“Coriolis”) and pressure gradient force (“PGF”) (third panel). Longitude-height cross section (bottom panel) of
meridional turbulent stress convergence (shaded) with meridional velocity (black)
and boundary layer depth (gray) contours overlaid.
105
FIGURE 4.5. Composites of surface pressure gradient force and SST. Top panel
shows zonal and bottom panel meridional maps of shaded “PGF” and contour
SST overlaid. Zero contour lines are omitted for clarity.
106
FIGURE 4.6. Composites of surface pressure gradient force and SST. Top panel
shows modeled PGF from the WRF simulation and bottom panel PGF estimates
based on the horizontal change of potential temperature θ (Eqn. 4.3). Contours
of SST are overlaid. Zero contour lines are omitted for clarity.
107
FIGURE 4.7. Boundary layer integral of zonal PGF and SST. The composite of
the top row is identical to the bottom row shifted by one-half wavelength to show
the spatial structure. Note that some additional smoothing has been applied to
remove very high wavenumber noise of unknown origin.
108
FIGURE 4.8. As in (Fig. 4.7, except for the zonal stress.
109
FIGURE 4.9. As in (Fig. 4.7, except for the zonal advection.
110
FIGURE 4.10. As in (Fig. 4.7, except for the zonal momentum budget.
111
5. A LINEAR DIAGNOSTIC MODEL OF ABL RESPONSE TO
MESOSCALE SST ANOMALIES
We now return to the central question of this thesis: What are the mechanisms responsible for the satellite-observed coupling between the surface wind
stress and SST anomalies with spatial scales of hundreds of kilometers? In Chapters 3 and 4 we have seen that vertical momentum exchange within the boundary
layer plays an important role in determining the internal structure of the boundary
layer. As pointed out by Samelson et al. (2006), however, these internal mechanisms can obscure the true nature of the coupling at the surface, where integral
constraints must be satisfied over the entire boundary layer.
5.1. A Linearized ABL-averaged diagnostic model
The results from Chapter 4 suggest that the vertically integrated response
of the ABL flow to SST anomalies may be represented by a simple linearized 1layer diagnostic model for perturbations of a purely zonal mean flow. A linearized
model allows us to provide an analytical expression for the surface wind speed
perturbations in terms of the SST perturbation.
The vertically integrated heat and momentum budgets are well represented
by:
∂
C θ U◦
θ=
(θsfc − θ◦ )
∂x
H
∂
1 ∂
U u=−
p − τx,◦ /H + f v
∂x
ρ◦◦ ∂x
1 ∂
∂
U v=−
p − τy,◦ /H − f u
∂x
ρ◦◦ ∂y
U
(5.1)
(5.2)
(5.3)
In Eqns. 5.1 – 5.3, U represents the vertically integrated mean zonal wind speed, u,
v ,p and θ represent vertically integrated perturbation quantities for wind, pressure
112
and potential temperature. Subscripts “◦” refer to surface ABL quantities, while
θsfc is the SST anomaly. Note that the perturbation wind does not enter the heat
budget since Cθ |u◦ |∆θ Cθ U◦ θsfc . H is the depth of the vertical integration,
which is assumed to be a constant level of zero stress.
The perturbation of θ can be related to the pressure gradient by vertically
integrating the hydrostatic equation to the height H where, in addition to being the
level of zero stress, we assume that the perturbation of the potential temperature
vanishes. This thermal boundary layer height Hθ is different from the momentum
BL height H in previous chapters. However, we assume that the two definitions
can be used interchangeably here because the numerical values are similar. The
surface pressure gradient then becomes:
−
Hθ
1
∇psfc = g ∇θ
ρ◦◦
θ◦◦
(5.4)
The analysis of the WRF-model suggests a simple linear decrease of the perturbation pressure gradient with increasing height (compare Fig. 5.1). It is then possible
to relate Eqn. 5.4 to the vertically integrated perturbation pressure gradient:
1 1
−
Hθ ρ◦◦
Z
Hθ
1 1 Hθ
∇psfc
Hθ ρ◦◦ 2
g Hθ
∇θ
=
2 θ◦◦
Γ
= ∇θ
2
∇p dz = −
0
(5.5)
(5.6)
(5.7)
Next, we replace the pressure gradient in Eqns. 5.2–5.3 with the ABL averaged
potential temperature gradient (Eqn. 5.7) and apply a simple drag law for the
surface stress, namely:
τx,◦ = Cd U◦ usfc = Cd U◦ ru u
(5.8)
τy,◦ = Cd U◦ vsfc = Cd U◦ rv v
(5.9)
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The coefficients ru,v represent a simple form of the shape function that relates
surface wind speed to the ABL-average velocity components. The choice of different values for the ratio between surface velocity and ABL-average velocity is
justified by the analysis of the WRF-model run in Chapter 4 where it was found
that the zonal ratio is several times larger than the meridional (Fig. 5.4) surface
to ABL-average velocity. This difference is related to differences in vertical shear
between the zonal and the meridional wind components (Figs. 4.3, 4.4). The
surface potential temperature θ◦ is related to the vertically averaged θ by:
θ◦ = rθ θ
(5.10)
The resulting three equations provide a complete set that can be solved
analytically for the three unknowns u, v and θ:
∂
θ=
∂x
∂
U u=
∂x
∂
U v=
∂x
U
C θ U◦
(θsfc − rθ θ)
Hθ
Γ ∂
θ − Cd U◦ ru u/Hθ + fv
2 ∂x
Γ ∂
θ − Cd U◦ rv v/Hθ − fu
2 ∂y
(5.11)
(5.12)
(5.13)
Note that the thermodynamic equation (Eqn. 5.11) is uncoupled from the
momentum equations (Eqns. 5.12 – 5.13). In other words,the feedback between
the heat flux and the perturbation wind is of secondary importance on the spatial
and temporal scales of interest in this study. Since the thermodynamic equation
(Eqn. 5.11) is uncoupled, the vertically averaged potential temperature θ (and
hence the pressure perturbation) can be obtained independently of the momentum
equations.
The exact analytical solutions of Eqns. 5.11–5.13 were obtained with the
help of Mathematica (see APPENDIX B). The solution for the homogeneous
part of the zonal and meridional momentum equation (Eqn. 5.12 and Eqn. 5.13)
114
involves two unknown integration “constants”. The inhomogeneous part is solved
by the technique known as variation of constants. The exact solution is provided
in Appendix B but it is instructive to obtain the polarization equation for the
system of equations. We assume a standard wave-like perturbation in the form
of:
θ = θ̃ · ei(kx+ly)
(5.14)
u = ũ · ei(kx+ly)
(5.15)
v = ṽ · ei(kx+ly)
(5.16)
and prescribe the SST perturbation as:
θsfc = θ̃sfc · ei(kx+ly)
(5.17)
The substitution of θ in Eqn. 5.11 with its wave form (Eqn. 5.14) can easily be
solved to yield the phase and gain relations between the potential temperature
amplitude θ̃ and the amplitude of the SST perturbation θ̃sfc . With the shorthand
notation of the inverse adjustment length scale:
a=
C θ U◦
Hθ · U
(5.18)
the thermodynamic equation is rewritten as:
θ̃ =
a
· θ̃sfc
rθ a + ik
(5.19)
and the phase ϕθ and gain Aθ can be calculated to yield the polarization relation
for the thermodynamic equation:
ϕθ = tan−1 (−
Aθ = p
a
rθ2 a2
k
)
rθ a
(5.20)
.
(5.21)
+ k2
115
The polarization relation between and ABL air potential temperature amplitude
θ̃ and SST θ̃sfc becomes:
a
tan−1 (−k/rθ a)
·
e
θ̃sfc
rθ2 a2 + k 2
θ̃ = p
(5.22)
The lack of a meridional derivative in Eqn. 5.11 indicates that the phase shift ϕθ
is purely zonal. The pattern of ABL potential temperature θ is therefore shifted
to the east relative to the pattern of SST by temperature advection, but remains
phase-locked with the SST perturbation in the meridional direction.
The values of the parameters are given in Tbl. 5.1. All parameters except
Cθ , U◦ and Hθ are determined by the model design: The horizontal scales k and
l and the SST perturbation θsfc are estimated from satellite observations, f and g
are physical constants. The vertical ratios r(u,v,θ) are determined from the WRF
simulation. The climatological zonal mean wind of about 10 m/s determines the
drag coefficient Cd .
The values for Cθ , U◦ and Hθ allowed more latitude for their specification
within physically reasonable limits. All three variables appear in Eqn. 5.11 and
therefore affect the amplitude and the phase of the perturbation air temperature
and hence the pressure. Fairly large heat exchange coefficient Cθ and surface
wind speed U◦ are necessary to simulate reasonably large pressure gradients. The
ABL depth H also influences magnitude of the pressure gradients. Moreover, Hθ
determines the phase shift of the surface wind to the SST perturbation, since Hθ
also appears in the momentum equations. The sensitivity of the model is nearly
linear in U◦ and Cθ in terms of the magnitude of the wind components for a
range of values of ± 50% of the optimal value. The phase relation remains nearly
unchanged. Varying the ABL depth Hθ has a lesser affect on wind speed magnitude than on the phase relation. Increasing Hθ shifts the zonal wind downwind
116
while decreasing the depth centers the zonal wind perturbation over the SST perturbation. The model seems robust over a wide range of values and it appears
that the stratification can influence the phase relation between wind and SST
perturbations.
Using the values from Tbl. 5.1, the numerical values for the zonal phase
ϕθ,x , meridional phase ϕθ,y and gain Aθ are:
ϕθ,x = −79.6◦
(5.23)
ϕθ,y = 0.◦
(5.24)
Aθ = 0.18
(5.25)
The ABL-averaged potential temperature perturbation is less than 20 % of the
SST perturbation. The thermodynamic adjustment length scale 1/a is much larger
than the length scale of the SST anomalies π/k. In other words, the time the air
spends over the anomaly is too short for air temperature to fully adjust. Therefore the sensible heat flux is roughly in phase with the SST anomaly. The heating associated with the SST perturbation is distributed over the depth of the
thermodynamic boundary layer Hθ . Therefore, even small potential temperature
perturbations can have significant pressure gradients.
Since the potential temperature and its zonal gradient are in quadrature,
and ϕθ,x = −79.6, the zonal potential temperature gradient is approximately in
phase with the patterns of SST; the meridional temperature gradient, however, is
90◦ out of phase with SST.
Pressure and air temperature are in phase. The zonal phase of the zonal
pressure gradient ϕ ∂p ,x and the meridional phase of the zonal pressure gradient
∂x
ϕ ∂p ,y are consequently almost zero relative to the pattern of SST:
∂x
117
ϕ ∂p ,x = 10.4◦
(5.26)
ϕ ∂p ,y = 0.◦
(5.27)
∂x
∂x
The meridional gradient operator introduces a 90◦ phase shift to the meridional
pressure gradient (Eqn. 5.13).
The meridional pressure gradient is therefore
shifted 90◦ to the north relative to the pattern of θ while it is in phase with
θ in the zonal direction. Numerically the phases are estimated to be:
ϕ ∂p ,x = −79.6◦
(5.28)
ϕ ∂p ,y = 90.◦
(5.29)
∂y
∂y
The amplitudes of the zonal and meridional pressure gradients are:
A ∂p = 4.1 · 10−5 ms−2 /K
(5.30)
∂x
A ∂p = 2.2 · 10−5 ms−2 /K
(5.31)
∂y
The polarization relations between ABL velocities and SST are estimated
from the exact solution (see APPENDIX B). The numerical estimate for the zonal
and meridional phase (ϕu,x and ϕu,y , respectively) and gain Au relative to the SST
perturbations are:
ϕu,x = −23.6◦
(5.32)
ϕu,y = 5.9◦
(5.33)
Au = 0.113 ms−1 /K
(5.34)
For the meridional velocity, the phases ϕv,(x,y) and gain Av are estimated to be:
ϕv,x = −141.6◦
(5.35)
ϕv,y = 81.4◦
(5.36)
Av = 0.095 ms−1 /K
(5.37)
118
5.2. Discussion
The comparisons between the 1-layer model surface wind fields and the
WRF surface wind fields in Figs. 5.2 – 5.3 demonstrate the ability of the 1-layer
model to capture the important features of the WRF simulation. The 1-layer
model zonal velocity (Fig. 5.2) shows an identical eastward and small southward
shift relative to the SST and the magnitude of surface wind perturbation is well
simulated. The meridional velocity perturbations in the WRF-model form continuous anomaly bands with a northwest-southeast orientation. The tendency for the
northwest-southeast banding on the scale of the SST anomalies is present in the
1-layer model as well, but the bands are less developed (Fig. 5.3). The response
of the surface meridional wind is shifted a bit further north in the WRF-model
run, which is likely related to the neglect of the meridional advection term in the
1-layer model and nonlinear stress effects.
The coupling between the surface wind stress and SST perturbations is
controlled by the pressure gradient force, and the primary dynamical balance is
between the pressure gradient force and the surface drag. The linearized zonal
∂
∂
u and U ∂x
v is responsible for the eastward shift of
advection of momentum U ∂x
the velocities relative to the pattern of SST. The Coriolis force plays a minor
role in the adjustment, but is responsible for the small meridional phase shift
of the zonal velocity ϕu,y of 5.9◦ . The large zonal phase shift of the meridional
velocity ϕv,x of 141.6◦ can be explained by considering the smaller gain of the
meridional pressure gradient A ∂p of 2.2 · 10−5 ms−2 /K compared to the gain of the
∂y
zonal pressure gradient, which is almost twice as large (Eqns. 5.30 and 5.31). The
meridional pressure gradient is therefore unable to balance the surface drag alone
and the phase shift from momentum advection becomes more pronounced.
119
Within the 1-layer model framework, the difference in coupling coefficients
between divergence and curl of the surface wind as a function of down and crosswind SST gradient can be explained by investigating the terms of the divergence
and vorticity budget. The vertically integrated budgets are readily derived from
Eqns. 5.12 – 5.13. As previously shown, the coefficient for the divergence-SST
coupling is close to twice as large as the coefficient for the curl-SST coupling. The
divergence budget is:
Γ 2
Cd U◦
∂u
∂v
∂
ru
+ rv
+fζ
U (∇ · v) = ∇ θ −
∂x
2
Hθ
∂x
∂y
∂v ∂u
where ζ =
−
is the vorticity
∂x ∂y
(5.38)
(5.39)
The largest term is the Laplacian of the pressure term (first term on the RHS,
equivalent to the Laplacian of the temperature). The pressure term is balanced
by the divergence of surface drag (second term on the RHS) and by the advection
of divergence (the LHS). The last term in Eqn. 5.38 is much smaller and can be
neglected. The vorticity budget is:
∂
Cd U◦
∂v
∂u
U ζ=
− ru
rv
+ f (∇ · v)
∂x
Hθ
∂x
∂y
(5.40)
The pressure term vanishes in the vorticity budget and the only remaining forcing
term is the divergence term (last term RHS), which is much smaller than the
pressure term in the divergence budget. The divergence in the vorticity budget
(Eqn. 5.40) is balanced by the advection of vorticity and the curl of surface drag.
All terms in the vorticity equation are much smaller than the leading terms in the
divergence equation. The pattern of surface divergence from the 1-layer solution is
shown in Fig. 5.5 and the pattern of the surface vorticity is shown in Fig. 5.6. The
two figures show that divergence is several times larger than vorticity and that
the pattern of divergence and vorticity are in quadrature to the pattern of SST.
The magnitudes and locations are in good agreement with the WRF simulations.
120
The difference between the zonal and the meridional ratio of surface to
ABL-average wind is important for the location of the vorticity relative to the
pattern of SST. This can be illustrated by assuming identical ratios (ru = rv = r)
and by rewriting the vorticity equation as
U
Cd U◦ r
∂
ζ=
ζ + f (∇ · v)
∂x
Hθ
(5.41)
In this form, the first term on the RHS provides only a damping to the solution while the LHS shifts the vorticity zonally. The vorticity should therefore be
damped and zonally shifted relative to the divergence pattern. However, observations show that the vorticity is located in quadrature to the pattern of divergence.
The meridional shift that moves the pattern into quadrature is due to the term
−ru ∂u
in Eqn. 5.40. The vertical structure of the ABL wind perturbations, as rep∂y
resented by the ABL-average wind ratios for the zonal (ru ) and meridional (rv )
direction, is therefore crucial for explaining the observed curl of the surface wind.
The good agreement between the 1-layer model and the WRF simulation can be
summarized by the scatter plots that quantify the relation between surface wind
curl and divergence as a function of SST gradient and by comparing the coupling
coefficients (see Eqns. 2.34 – 2.35) to previous results (e.g. Chelton et al. 2004,
O’Neill 2003 and 2005). The coupling coefficients for the divergence and curl as
functions of downwind and crosswind SST gradients (Fig. 5.7) are similar to the
results from Chapter 1. The amplitudes (Fig. 5.8) of the divergence and curl as
functions of the angle between components of downwind and crosswind SST gradient are larger than in the WRF simulation, but the tendency that the amplitude
of the divergence is about two times larger than the curl amplitude remains. The
coupling coefficient for the divergence as a function of downwind SST gradient
can be directly estimated from the gain in the polarization relation (Eqn. 5.34).
121
Since the zonal velocity amplitude is larger than the amplitude of the meridional
velocity, and because the zonal wave lengths are smaller than the meridional we
can approximate the divergence by the gradient of the zonal velocity. Ignoring
the phase shift (Eqns. 5.32 – 5.33), the polarization relation for the zonal surface
velocity can be written:
ũsfc = ru · Au SST
(5.42)
(5.43)
The differentiation is approximated by a finite difference ∆x:
∆SST
∆ũsfc
= 6. · 0.113
∆x
∆x
∆ũsfc
∆SST
= 0.678
∆x
∆x
(5.44)
(5.45)
The value of 0.678 is identical to the coupling coefficient obtained from the binned
scatter plots (Fig. 5.7).
Although pressure perturbations are the driving force in this adjustment
process, the simple 1-layer model has shown that the vertical structure of the wind
response, which is expressed here in simple surface/ABL-ratio, is very important.
The vertical shear allows the generation of larger vorticity than with vertical
uniform wind and is responsible for shifting the vorticity over the regions with
large crosswind SST gradients.
The vertical structure is only crudely represented in this 1-layer model
and does not explain the differences in the vertical structure of the downwind
and crosswind wind perturbation components. The 1-layer model may also be
inadequate for investigating the scale dependent response of the surface stress to
SST anomalies in the vicinity of SST fronts. This will require the inclusion of
mixing and adjustment processes similar to the ones described in Chapter 3, and
is left for future work.
122
Parameter:
Value:
Sensitivity
k
2 · π/289 km−1
n/a
l
2 · π/532 km−1
n/a
f
−5 × 10−5 s−1
n/a
g
9.81 ms−2
n/a
θ◦
280 K
n/a
θsfc
2.5 K
n/a
U
10 ms−1
n/a
Cd
1.4×10−3
n/a
ru
6
n/a
rv
2
n/a
rθ
2
n/a
Cθ
2.5 × 10−3
large, effects PGF
U◦
4 ms−1
moderate, effects PGF
Hθ
500 m
small, phase shift
TABLE 5.1. Parameters for the 1-layer linear perturbation model. The first 11
parameters were pre-determined by the experimental design and physics. The last
three values represent model parameters with a range of physically correct values
(see text for details). For these three values the optimal value within the physical
range were chosen. The sensitivity of the model is indicated and the influence
effect is given.
123
FIGURE 5.1. Latitude-height cross section of estimated pressure gradient force
with ABL depth (bottom panel) and SST (top panel). Estimate of “PGF” is
shaded with contour of ABL depth (gray) as overlay.
124
FIGURE 5.2. Map of 1-layer diagnostic model zonal velocity (top) and
WRF-model averaged zonal surface velocity (repeated for comparison, bottom).
SST contours are shown as overlays with contour interval of .5◦ C. Zero contour
of SST is omitted for clarity.
125
FIGURE 5.3. Map of 1-layer diagnostic model meridional velocity (top) and
WRF-model averaged meridional surface velocity (repeated for comparison, bottom). SST contours are shown as overlays with contour interval of .5◦ C. Zero
contour of SST is omitted for clarity.
126
FIGURE 5.4. Map of surface to vertically integrated meridional velocity ratio
derived from WRF-model simulation. Top panel shows zonal velocity ratio, bottom panel shows meridional velocity ratio. Locations with vertically integrated
velocities of less than 0.05 m/s have been masked. SST contours are shown as
overlays with contour interval of .5◦ C. Positive values of SST is shown as solid and
negative SST as dashed contours, the zero contour of SST is omitted for clarity.
127
FIGURE 5.5. Map of 1-layer diagnostic model divergence and SST perturbations.
128
FIGURE 5.6. Map of 1-layer diagnostic model curl and SST perturbations.
129
Diagnostic Model Perturbation Divergence
2
1.5
m/s per 100km
1
0.5
0
−0.5
−1
−1.5
Slope=0.68
−2
−2
−1.5
−1
−0.5
0
0.5
1
Perturbation Downwind SST Gradient (°C per 100km)
1.5
2
Diagnostic Model Perturbation Curl
2
1.5
m/s per 100km
1
0.5
0
−0.5
−1
−1.5
Slope=−0.42
−2
−2
−1.5
−1
−0.5
0
0.5
1
Perturbation Crosswind SST Gradient (°C per 100km)
1.5
2
FIGURE 5.7. Binned scatterplots of the relationships between the SST and wind
velocity fields calculated from the diagnostic solution of the 1-layer model. Top
panel shows perturbation wind divergence as a function of downwind SST gradient
and bottom panel shows perturbation wind curl plotted as a function of crosswind
SST gradient.
130
Diagnostic Model Perturbation Divergence
2
1.5
m/s per 100km
1
0.5
0
−0.5
−1
−1.5
Amplitude=1.23
−2
−180
−135
−90
−45
0
θ′ (deg)
45
90
135
180
Diagnostic Model Perturbation Curl
2
1.5
m/s per 100km
1
0.5
0
−0.5
−1
−1.5
Amplitude=0.60
−2
−180
−135
−90
−45
0
θ′ (deg)
45
90
135
180
FIGURE 5.8. Binned scatterplots of the angular relationships between the wind
and the SST field. Top panel shows perturbation wind divergence as a function of
the angle between down and crosswind SST gradient and bottom panel shows perturbation wind curl plotted as a function of the angle between down and crosswind
SST gradient.
131
6. SUMMARY
This thesis has examined the mechanisms that couple the monthlyaveraged atmospheric boundary layer (ABL) to sea surface temperature (SST)
perturbations over the open ocean on scales of 50-500 km. Numerical and analytical methods were used to simulate and analyze the response of the surface wind
to SST perturbations. Numerical experiments were performed with the Weather
Research and Forecasting (WRF) mesoscale model to simulate the ABL response
to realistic and idealized SST anomalies. Detailed analysis of the simulated momentum budgets lead to the development of a 1-layer diagnostic analytical model
that successfully predicts the observed relationship between surface wind and SST
anomalies in the Agulhas return current region. The analytical model provides
increased understanding of the mechanisms of atmosphere-ocean coupling.
The first numerical experiment established the ability of the WRF model
to accurately simulate the surface wind response revealed by QuikSCAT satellite
observations over the Agulhas return current. We found that the WRF model
successfully simulates the observed correspondence between higher surface winds
and positive SST anomalies and between lower surface winds and negative SST
anomalies. The comparison with observations was further quantified by computing
the coupling coefficients between the surface divergence and surface curl in terms
of the downwind and crosswind SST gradients, respectively, and also by the phase
and amplitudes of the sinusoidal response of the surface divergence and surface curl
as a function of the counterclockwise angle between the downwind and crosswind
SST gradients. With the exception of the amplitude of the surface divergence as
a function of the counterclockwise angle, the quantitative agreement between the
WRF simulations and observations is very good.
132
Next, a pair of WRF numerical experiments focused on the response of
the atmospheric boundary layer to idealized SST fronts. Two independent model
runs were performed: (1) a cold-to-warm SST front integration (P4), and (2) a
warm-to-cold front integration (M4). The acceleration of the surface wind in P4
and deceleration in M4, were found over a narrow transition zone co-located with
the narrow region of largest SST changes. Horizontal momentum is redistributed
vertically in the ABL by turbulence and convection in the transition zone, while
largest pressure adjustments take place over a much broader region downstream
from the SST front. In the cold-to-warm transition (P4), WRF simulates an
unstable, growing thermal internal boundary layer (TIBL) in the lower part of
the ABL. As the TIBL grows, higher velocity air aloft is incorporated into the
TIBL. In the warm-to-cold case (M4), the momentum boundary layer collapses
over the transition zone, and vertical mixing of momentum by turbulence and
convection ceases in the upper part of the ABL. We conclude that high vertical
resolution is required to represent the important ABL adjustment mechanisms
associated with SST fronts within the ABL.
The final WRF experiment simulated the atmospheric response to idealized
mesoscale SST anomalies arranged in a checkerboard pattern over the southern
middle latitudes. The alternating positive and negative SST anomalies are assumed to have spatial scales similar to those observed in the Agulhas region. The
horizontal scale of the ABL pressure response was found to be similar to the scale
of the SST perturbation itself and the horizontal pressure gradient force plays an
important role in the vertically integrated horizontal momentum budget. Temperature advection by the mean wind shifts the perturbation pressure gradient
over the SST anomalies and a balance is found between the pressure gradient and
the perturbation stress convergence of the momentum boundary layer.
133
Analysis of the vertically integrated momentum budgets from the WRF
simulations showed that the ABL response to the Agulhas mesoscale SST anomalies was found to be approximately linear. A linear diagnostic model therefore
was developed and analyzed that improves our understanding of the mechanisms
coupling the surface wind to the SST anomalies. The analytical model successfully predicts the observed phase and amplitude of the ABL wind, pressure and
temperature response to the SST anomalies, with largest quantitative discrepancies found in the perturbation wind component perpendicular to the mean wind
direction. By using the divergence and vorticity budgets, the diagnostic model
shows how the difference of vertical shear between the cross and downwind wind
component might explain the differences in the coupling coefficients between the
divergence and curl as functions of downwind and crosswind SST gradients.
134
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139
APPENDICES
140
APPENDIX A. WRF-model η-levels and nominal heights
• η:
1.0000 0.9967 0.9932 0.9896 0.9858 0.9820 0.9779 0.9738 0.9694 0.9649 0.9603
0.9554 0.9504 0.9452 0.9398 0.9342 0.9284 0.9224 0.9161 0.9096 0.9029 0.8959
0.8886 0.8811 0.8733 0.8652 0.8568 0.8481 0.8391 0.8297 0.8200 0.8099 0.7994
0.7886 0.7773 0.7656 0.7535 0.7409 0.7279 0.7143 0.7003 0.6857 0.6706 0.6549
0.6386 0.6217 0.6042 0.5860 0.5672 0.5476 0.5273 0.5063 0.4845 0.4618 0.4383
0.4139 0.3886 0.3624 0.3351 0.3069 0.2776 0.2472 0.2157 0.1829 0.1490 0.1138
0.0772 0.0393 0.0000
• Heights in m
12.00 34.66 57.36 82.11 107.59 134.14 161.76 190.46 220.59 251.48 283.83
317.60 352.52 388.94 426.85 466.27 507.23 550.08 594.86 641.23 689.58 740.28
793.02 847.83 905.11 964.90 1027.24 1092.20 1160.18 1231.25 1305.49 1383.32
1464.46 1549.37 1638.54 1731.68 1829.29 1931.51 2038.88 2151.56 2269.7
2394.00 2524.58 2662.14 2806.97 2959.36 3120.10 3289.58 3468.71 3658.46
3858.93 4071.27 4297.24 4537.74 4793.81 5067.17 5359.21 5672.74 6009.72
6372.49 6765.15 7191.25 7656.76 8167.61 8731.80 9362.85 10076.89 10900.07
(A.2)
(k 2
+
a2 rθ2 )
H 4k4U 4
2 4
2
H U k + (f Hl + C◦ krv )
− H 2 k 2 (2f 2 H 2 − C◦2 (ru2 + rv2 )) U 2 + (ru rv C◦2 + f 2 H 2 )2
2
!1/2
(A.4)
H 2 k 2 l2 U 2 + (f Hk − C◦ lru )2
(k 2 + a2 rθ2 ) H 4 k 4 U 4 − H 2 k 2 (2f 2 H 2 − C◦2 (ru2 + rv2 )) U 2 + (ru rv C◦2 + f 2 H 2 )2
!1/2
(A.7)
(A.6)
HU (k 2 lU 2 H 2 − f 2 lH 2 − C◦ f k(ru + rv )H + C◦2 lru2 ) k 2 + arθ ((f Hk − C◦ lru ) (ru rv C◦2 + f 2 H 2 ) − H 2 k 2 (f Hk + C◦ lrv )U 2 )
(A.5)
k (f 3 kH 3 + f 2 l(aHrθ U − C◦ ru )H 2 + f k (ru rv C◦2 + aHrθ (ru + rv )U C◦ − H 2 k 2 U 2 ) H − l(C◦ rv + aHrθ U ) (C◦2 ru2 + H 2 k 2 U 2 ))
Av = aHΓ
ϕv =
a
+ a2 rθ2
(A.1)
arθ ((f Hl + C◦ krv ) (ru rv C◦2 + f 2 H 2 ) − H 2 k 2 (f Hl − C◦ kru )U 2 ) − Hk 2 U (H 2 U 2 k 3 − f 2 H 2 k + C◦2 rv2 k + C◦ f Hl(ru + rv ))
(A.3)
k (f 3 lH 3 + f 2 k(C◦ rv − aHrθ U )H 2 + f l (ru rv C◦2 + aHrθ (ru + rv )U C◦ − H 2 k 2 U 2 ) H + k(C◦ ru + aHrθ U ) (C◦2 rv2 + H 2 k 2 U 2 ))
Au = aHΓ
ϕu =
k2
k
arθ
Aθ = p
ϕθ = −
APPENDIX B. Solution to the Diagnostic Model Appendix
141
aC◦ rθ k 2 rv U 2 H 2 + f 2 ru H 2 + C◦2 ru2 rv − Hk 2 U
(A.9)
(A.8)
(A.10)
k 2 U 2 − f 2 H 2 + C◦2 ru2 cos(kx)
+k ru2 rv C◦3 + aHrθ ru2 U C◦2 + H 2 ru f 2 + k 2 rv U 2 C◦ + aH 3 rθ U k 2 U 2 − f 2 sin(kx)
− f Hk k ru rv C◦2 + H(aC◦ rθ (ru + rv )U + H(f − kU )(f + kU )) cos(kx)
+ C◦ Hk 2 (ru + rv )U − arθ ru rv C◦2 + H 2 (f − kU )(f + kU ) sin(kx) sin(ly) /
h
i
2
4 4 4
2
2
2 2 2
2
2
2 2
2 2
2 2
2
k + a rθ H k U − H k 2f H − C◦ ru + rv U + ru rv C◦ + f H
v(x, y) =
h
aH θ̃sfc Γ l cos(ly)
+k ru rv C◦2 + H(aC◦ rθ (ru + rv )U + H(f − kU )(f + kU )) sin(kx)
+ k k ru rv2 C◦3 + aHrθ rv2 U C◦2 + H 2 rv f 2 + k 2 ru U 2 C◦ + aH 3 rθ U k 2 U 2 − f 2 cos(kx)
− aC◦ rθ k 2 ru U 2 H 2 + f 2 rv H 2 + C◦2 ru rv2 − Hk 2 U k 2 U 2 − f 2 H 2 + C◦2 rv2 sin(kx) sin(ly) /
h
2 i
k 2 + a2 rθ2 H 4 k 4 U 4 − H 2 k 2 2f 2 H 2 − C◦2 ru2 + rv2 U 2 + ru rv C◦2 + f 2 H 2
arθ ru rv C◦2 + H 2 (f − kU )(f + kU ) − C◦ Hk 2 (ru + rv )U cos(kx)
aθ̃sfc (arθ cos(kx) + k sin(kx)) sin(ly)
k 2 + a2 rθ2
u(x, y) =
h
aH θ̃sfc Γ f Hl cos(ly)
θ(x, y) =
142